#7623357

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623305
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
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1696 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)(11,21)(12,20)(13,16)(14,27)(15,18)(17,24)(19,29)(22,37)(23,39)(25,32)(26,34)(28,36)(30,35)(31,38)(33,40);(1,3,4,8)(2,5,6,7)(10,16)(11,22)(18,20)(19,32)(23,30)(24,38)(27,34)(28,40);(3,8)(5,7)(9,11)(10,17)(12,23)(13,22)(14,28)(15,30)(16,31)(18,33)(19,35)(20,36)(21,27)(24,29)(25,38)(26,40)(32,39)(34,37);(2,6)(5,7)(9,12,13,15)(10,18,16,20)(11,23,22,30)(14,25,26,29)(17,33,31,36)(19,34,32,27)(21,35,37,39)(24,28,38,40);(10,16)(17,31)(18,20)(19,32)(21,37)(27,34)(33,36)(35,39)

Polycyclic presentation:

2^9|b^2=fi,d^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gh,h^a=hi,c^b=cf,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530553);