#7622764

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁵
commutator, socle	C₂²	C₂⁴×C₈
prime radical	#3483	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1341 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)(3,7)(4,6)(5,8)(9,10,13,18,14,19,26,32,15,12,17,24,27,25,31,37)(11,16,21,28,22,29,34,38,20,23,30,35,33,36,39,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,23)(13,21)(14,22)(15,20)(17,30)(18,28)(19,29)(24,35)(25,36)(26,34)(27,33)(31,39)(32,38)(37,40);(3,8)(5,7)(9,11)(10,16)(12,23)(13,21)(14,22)(15,20)(17,30)(18,28)(19,29)(24,35)(25,36)(26,34)(27,33)(31,39)(32,38)(37,40);(2,6)(5,7)(11,20)(16,23)(21,30)(22,33)(28,35)(29,36)(34,39)(38,40);(9,12,13,24,14,25,26,37,15,10,17,18,27,19,31,32)(11,16,21,28,22,29,34,38,20,23,30,35,33,36,39,40)

Polycyclic presentation:

2^9|a^2=g,e^2=g,g^2=h,h^2=i,b^a=bf,d^a=df,c^b=cf,d^b=di,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529960);