#6671065

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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1804 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)(5,6,8,9)(7,11,12,10)(13,14,18,22)(15,19,25,33)(16,20,27,35)(17,21,29,37)(23,31,30,38)(24,32,40,43)(26,34,28,36)(39,42,41,44);(5,6)(7,10)(8,9)(11,12)(15,23)(16,26)(17,29)(19,31)(20,34)(21,37)(24,40)(25,30)(27,28)(32,43)(33,38)(35,36);(2,4)(6,9)(10,11)(13,15,18,25)(14,19,22,33)(16,24,27,40)(17,30,29,23)(20,32,35,43)(21,38,37,31)(26,41,28,39)(34,44,36,42);(5,7,8,12)(6,10,9,11)(16,27)(20,35)(24,40)(26,28)(32,43)(34,36)(39,41)(42,44);(13,16,18,27)(14,20,22,35)(15,24,25,40)(17,26,29,28)(19,32,33,43)(21,34,37,36)(23,39,30,41)(31,42,38,44)

Polycyclic presentation:

2^9|a^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=efhi,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, 6578261);