#72319391

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	#7623099	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

2470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|c^2=hj,d^2=f,f^2=h,g^2=ij,i^2=j,c^a=cj,c^b=cg,d^c=dhi,e^b=eh,e^d=ehj,f^c=fj,g^b=gi,g^c=gi,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	#72319510, #72319579, #72319621, …
maximal normal subgroup of	#72319510, #72319579, #72319621, …

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