#6671073

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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1797 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)(13,14,19,18)(15,20,25,31)(16,21,28,32)(17,22,30,33)(23,34,38,26)(24,29,36,39)(27,35,42,43)(37,44,41,40);(5,6)(7,10)(8,9)(11,12)(13,15,19,25)(14,20,18,31)(16,26,28,34)(17,29,30,39)(21,23,32,38)(22,36,33,24)(27,41,42,37)(35,40,43,44);(2,4)(6,9)(10,11)(13,16,19,28)(14,21,18,32)(15,23,25,38)(17,27,30,42)(20,34,31,26)(22,35,33,43)(24,37,36,41)(29,44,39,40);(5,7,8,12)(6,10,9,11)(17,30)(22,33)(24,36)(27,42)(29,39)(35,43)(37,41)(40,44);(13,17,19,30)(14,22,18,33)(15,24,25,36)(16,27,28,42)(20,29,31,39)(21,35,32,43)(23,37,38,41)(26,40,34,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,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, 6578269);