#6809170

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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×#3282	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12);(6,9)(7,11)(13,14)(15,24)(16,21)(17,30)(18,34)(19,23)(20,25)(22,32)(26,38)(27,35)(28,37)(29,39)(31,36)(33,44)(40,41)(42,43);(2,4)(13,15,18,25,19,27,16,28)(14,20,21,24,23,37,34,35)(17,31,32,42,33,26,29,41)(22,38,30,43,39,36,44,40);(6,9)(10,12)(13,16,19,18)(14,21,23,34)(15,25,27,28)(17,29,33,32)(20,35,37,24)(22,30,39,44)(26,41,31,42)(36,43,38,40);(5,7,8,11)(6,10,9,12)(13,17,18,32,19,33,16,29)(14,22,21,30,23,39,34,44)(15,26,25,41,27,31,28,42)(20,36,24,40,37,38,35,43)

Polycyclic presentation:

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

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, 6716366);