#72322786

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	#10575373	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ C₂×C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4917 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:

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

Polycyclic presentation:

2^10|b^2=f,d^2=h,f^2=h,g^2=ij,i^2=j,b^a=bg,c^a=ch,d^a=dhj,e^d=ej,f^a=fj,g^a=gi,g^b=gij,i^a=ij,i^b=ij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322814
maximal normal subgroup of	#72322814

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