#6809177

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×#3284	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2259 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)(15,20,25,34)(16,21,29,27)(17,22,33,32)(23,26,35,39)(24,30,37,44)(28,36,31,38)(40,42,41,43);(6,9)(7,11)(13,15)(14,20)(16,26)(17,30)(18,34)(19,25)(21,35)(22,37)(23,27)(24,32)(28,41)(29,39)(31,40)(33,44)(36,43)(38,42);(2,4)(13,16,18,27,19,29,14,21)(15,23,20,26,25,35,34,39)(17,31,32,36,33,28,22,38)(24,41,30,43,37,40,44,42);(6,9)(10,12)(13,14,19,18)(15,20,25,34)(16,27,29,21)(17,22,33,32)(23,39,35,26)(24,30,37,44)(28,38,31,36)(40,43,41,42);(5,7,8,11)(6,10,9,12)(13,17,18,32,19,33,14,22)(15,24,20,30,25,37,34,44)(16,28,27,38,29,31,21,36)(23,40,26,42,35,41,39,43)

Polycyclic presentation:

2^9|a^2=i,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, 6716373);