#6996581

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6903777);