#6669131

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2045 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)(5,6)(7,10)(8,9)(11,12)(13,14,19,18)(15,20,25,32)(16,21,28,33)(17,29,31,22)(23,34,38,26)(24,30,39,35)(27,41,42,36)(37,40,44,43);(6,9)(7,11)(13,15,19,25)(14,20,18,32)(16,26,28,34)(17,30,31,35)(21,23,33,38)(22,24,29,39)(27,37,42,44)(36,43,41,40);(2,4)(5,7,8,11)(6,10,9,12)(13,16,19,28)(14,21,18,33)(15,23,25,38)(17,27,31,42)(20,34,32,26)(22,36,29,41)(24,37,39,44)(30,40,35,43);(6,9)(10,12)(13,14,19,18)(15,20,25,32)(16,21,28,33)(17,29,31,22)(23,34,38,26)(24,30,39,35)(27,41,42,36)(37,40,44,43);(13,17,19,31)(14,22,18,29)(15,24,25,39)(16,27,28,42)(20,35,32,30)(21,36,33,41)(23,37,38,44)(26,40,34,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6576327);