#7622658

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂⁴
commutator, socle	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#3444	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1394 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)(3,6)(4,5)(7,8)(9,10,12,17,13,18,22,30,14,16,23,28,24,29,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40);(2,5)(6,8)(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40);(9,11,12,20,13,21,22,33,14,19,23,31,24,32,34,39)(10,15,17,26,18,27,30,37,16,25,28,35,29,36,38,40);(1,3,4,7)(2,6,5,8);(10,16)(15,25)(17,28)(18,29)(26,35)(27,36)(30,38)(37,40)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=f,g^2=h,h^2=i,b^a=bf,e^a=ei,c^b=ci

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, 7529854);