#7622727

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	#3475	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1310 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=f,g^2=h,h^2=i,b^a=bf,c^a=ci,d^a=df,c^b=cfi,e^d=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, 7529923);