#6807023

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×#3197	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2049 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6714219);