#6808006

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1808 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)(13,14,19,18)(15,20,25,34)(16,21,27,29)(17,30,33,22)(23,26,35,42)(24,40,37,31)(28,38,32,36)(39,44,41,43);(5,6)(7,10)(8,9)(11,12)(13,15,19,25)(14,20,18,34)(16,26,27,42)(17,31,33,40)(21,35,29,23)(22,37,30,24)(28,41,32,39)(36,44,38,43);(2,4)(13,16,14,21,19,27,18,29)(15,23,34,42,25,35,20,26)(17,32,22,38,33,28,30,36)(24,41,40,43,37,39,31,44);(5,7,8,11)(6,10,9,12)(16,27)(21,29)(23,35)(26,42)(28,32)(36,38)(39,41)(43,44);(6,9)(10,12)(13,17,14,22,19,33,18,30)(15,24,34,40,25,37,20,31)(16,28,21,36,27,32,29,38)(23,39,42,44,35,41,26,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=f,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6715202);