#7622769

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	#3460	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1332 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)(9,10,13,17,14,18,25,31,12,15,23,26,24,27,36,37)(11,19,21,32,22,33,35,40,20,16,28,29,34,30,38,39);(1,3)(2,5)(4,8)(6,7)(10,15)(16,19)(17,26)(18,27)(29,32)(30,33)(31,37)(39,40);(3,8)(5,7)(10,15)(16,19)(17,26)(18,27)(29,32)(30,33)(31,37)(39,40);(2,6)(5,7);(9,11,13,21,14,22,25,35,12,20,23,28,24,34,36,38)(10,16,17,29,18,30,31,39,15,19,26,32,27,33,37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320856, #72322022, #72322106
maximal normal subgroup of	#72320856, #72322022, #72322106

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7529965);