#7149156

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1874 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056352);