#6814410

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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

1908 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6721606);