#7624014

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×#509
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gh,h^a=hi,c^b=ci,d^b=df,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, 7531210);