#7623176

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622976
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#60643	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1642 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)(11,23)(12,30)(13,19)(14,21)(15,20)(16,33)(17,22)(18,27)(24,26)(25,36)(28,38)(29,39)(31,32)(34,40)(35,37);(2,5)(6,8)(9,11,15,28,16,29,32,37,17,26,30,40,13,25,14,27)(10,18,21,36,19,34,12,24,22,35,31,39,33,38,20,23);(1,3,4,7)(2,6,5,8)(9,12,14,19,13,21,30,10,17,20,32,33,16,31,15,22)(11,24,27,34,25,36,40,18,26,23,37,38,29,39,28,35);(9,13,17,16)(10,19,22,33)(11,25,26,29)(12,31,20,21)(14,30,32,15)(18,34,35,38)(23,36,24,39)(27,40,37,28);(11,26)(18,35)(23,24)(25,29)(27,37)(28,40)(34,38)(36,39)

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=i,f^2=hi,g^2=h,h^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=ghi,h^a=hi,e^b=ei,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530372);