#6836107

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6743303);