#6716730

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1312 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)(3,6)(4,8)(5,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,28)(27,31,36,37)(29,32,39,38)(33,35,40,34);(1,3,5,4)(2,6,7,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,31)(28,37)(29,34)(30,36)(32,33)(35,39)(38,40);(2,7)(6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,26,30,28)(27,33,36,40)(29,38,39,32)(31,35,37,34);(10,16)(14,17)(15,22)(21,23)(25,28,30,26)(27,31,36,37)(29,38,39,32)(33,35,40,34);(1,4,5,3)(2,8,7,6)(11,19)(14,17)(18,24)(21,23)(27,34)(29,39)(31,33)(32,38)(35,36)(37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fi,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,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, 6623926);