#72318760

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#10580937	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4567 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:

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

Polycyclic presentation:

2^10|b^2=f,f^2=h,g^2=i,i^2=j,b^a=bhj,c^a=ch,c^b=cg,d^a=dj,e^b=ej,e^d=ehj,f^c=fj,g^b=gi,g^c=gij,i^b=ij,i^c=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318841, #72319144, #72319749
maximal normal subgroup of	#72318841, #72319144, #72319749

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