#6816133

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

2178 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,e^2=fh,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, 6723329);