#6807973

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#42652	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6715169);