#6807153

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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2011 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,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, 6714349);