#72318667

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, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623517	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2281 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:

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

Polycyclic presentation:

2^10|b^2=f,d^2=fhj,f^2=h,g^2=i,i^2=j,c^b=cg,d^a=dh,d^b=dj,d^c=di,e^b=eh,e^c=ehj,e^d=ehj,f^c=fj,g^b=gi,g^c=gij,g^d=gj,i^b=ij,i^c=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318707, #72319144, #72319265
maximal normal subgroup of	#72318707, #72319144, #72319265

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