#6811299

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	#25390	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1857 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6718495);