#7622757

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	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂⁴×C₈
Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#3459	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1342 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)(9,10,12,17,13,18,22,30,14,15,23,26,24,27,34,37)(11,16,19,28,20,29,31,38,21,25,32,35,33,36,39,40);(1,3)(2,5)(4,8)(6,7);(3,8)(5,7)(10,15)(16,25)(17,26)(18,27)(28,35)(29,36)(30,37)(38,40);(2,6)(5,7);(9,11,12,19,13,20,22,31,14,21,23,32,24,33,34,39)(10,16,17,28,18,29,30,38,15,25,26,35,27,36,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322106
maximal normal subgroup of	#72322106

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529953);