#72319231

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁵
prime radical	#6491106	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,6,7)(3,11,13,8)(4,10,5,9)(12,15,14,16)(17,18,22,23)(19,27,30,24)(20,31)(21,35,37,26)(25,34)(28,42)(29,38,46,43)(32,39)(33,41)(36,40)(44,48)(45,47);(2,7)(8,11)(9,10)(15,16)(17,19)(18,24)(20,32,34,42)(21,29)(22,30)(23,27)(25,39,31,28)(26,38)(33,48,36,47)(35,43)(37,46)(40,44,41,45);(1,3)(2,8)(4,14)(5,12)(6,13)(7,11)(9,16)(10,15)(17,20)(18,25,23,31)(19,28,30,39)(21,36,37,33)(22,34)(24,32)(26,41)(27,42)(29,45)(35,40)(38,48,43,47)(44,46);(1,4,6,5)(2,9,7,10)(3,12,13,14)(8,15,11,16)(17,21,22,37)(18,26,23,35)(19,29,30,46)(20,33,34,36)(24,38,27,43)(25,40,31,41)(28,44,39,45)(32,47,42,48);(1,5,6,4)(2,10,7,9)(3,12,13,14)(8,15,11,16)

Polycyclic presentation:

2^10|a^2=f,b^2=i,c^2=h,d^2=fi,e^2=fij,g^2=j,b^a=bfi,c^a=cfh,c^b=cg,d^a=df,d^b=di,d^c=dfj,e^a=efij,f^c=fj,g^a=gi,g^c=gi,g^d=gj,h^a=hj,h^b=hij,h^d=hj,h^g=hj,i^c=ij

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319313
maximal normal subgroup of	#72319313

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.6*F.9);;
gap> SetPower(coll, 5, F.6*F.9*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);