#7622778

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
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	#3460	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1305 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

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,e^c=ei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529974);