#6671689

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2184 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)(15,20,25,31)(16,21,28,32)(17,22,30,33)(23,34,38,26)(24,35,39,29)(27,36,41,42)(37,43,44,40);(5,6,8,9)(7,10,11,12)(13,15,19,25)(14,20,18,31)(16,26,28,34)(17,29,30,35)(21,23,32,38)(22,24,33,39)(27,37,41,44)(36,43,42,40);(2,4)(13,16,19,28)(14,21,18,32)(15,23,25,38)(17,27,30,41)(20,34,31,26)(22,36,33,42)(24,37,39,44)(29,40,35,43);(5,7,8,11)(6,10,9,12)(17,30)(22,33)(24,39)(27,41)(29,35)(36,42)(37,44)(40,43);(6,9)(10,12)(13,17,19,30)(14,22,18,33)(15,24,25,39)(16,27,28,41)(20,35,31,29)(21,36,32,42)(23,37,38,44)(26,40,34,43)

Polycyclic presentation:

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

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, 6578885);