#72318641

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂²×#442
prime radical	#7623096	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2506 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)(3,8)(4,9)(5,15)(6,11)(7,12)(10,14)(13,16)(17,18)(19,23)(20,26)(21,25)(22,27)(24,37)(28,38)(29,46)(30,39)(31,40)(32,41)(33,42)(34,45)(35,44)(36,43)(47,48);(1,3,6,5,7,13,4,14)(2,8,11,15,12,16,9,10)(17,19,21,31,22,30,35,33)(18,23,25,40,27,39,44,42)(20,29,36,32,37,28,34,47)(24,38,45,48,26,46,43,41);(1,4,7,6)(2,9,12,11)(3,5,13,14)(8,15,16,10)(19,28)(20,34)(21,35)(23,38)(24,43)(25,44)(26,45)(29,30)(31,32)(33,47)(36,37)(39,46)(40,41)(42,48);(1,5,6,13,7,14,4,3)(2,10,11,8,12,15,9,16)(17,20,35,34,22,37,21,36)(18,24,44,43,27,26,25,45)(19,29,33,47,30,28,31,32)(23,38,42,41,39,46,40,48);(19,30)(23,39)(28,29)(31,33)(32,47)(38,46)(40,42)(41,48)

Polycyclic presentation:

2^10|b^2=f,c^2=hj,d^2=fj,f^2=h,g^2=i,i^2=j,c^b=cg,d^a=dh,d^b=dhj,d^c=dhi,e^b=ej,f^c=fj,g^b=gi,g^c=gij,g^d=gj,i^b=ij,i^c=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318707, #72319199, #72319510
maximal normal subgroup of	#72318707, #72319199, #72319510

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, 2, F.6);;
gap> SetPower(coll, 3, F.8*F.10);;
gap> SetPower(coll, 4, F.6*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);