#6671057

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1757 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);(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,23,26,38)(16,27,28,37)(17,30,33,21)(19,31,35,25)(20,29,36,32)(24,40,41,34)(39,42,44,43);(2,4)(6,9)(10,11)(13,15,18,26)(14,19,22,35)(16,24,28,41)(17,31,33,25)(20,34,36,40)(21,38,30,23)(27,44,37,39)(29,43,32,42);(5,7,8,12)(6,10,9,11)(16,28)(20,36)(24,41)(27,37)(29,32)(34,40)(39,44)(42,43);(13,16,18,28)(14,20,22,36)(15,24,26,41)(17,32,33,29)(19,34,35,40)(21,27,30,37)(23,39,38,44)(25,43,31,42)

Polycyclic presentation:

2^9|a^2=f,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, 6578253);