#7503962

order	512
nilpotency class	3
isoclinism	isoclinic class of #7501009
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1789 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)(3,4)(5,7)(6,8)(9,10,15,17,14,19,13,20)(11,16,23,32,24,30,21,31)(12,25,27,34,28,33,26,18)(22,36,35,39,38,40,37,29);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,22)(13,21)(14,24)(15,23)(17,32)(18,29)(19,30)(20,31)(25,36)(26,37)(27,35)(28,38)(33,40)(34,39);(10,17)(11,21)(13,15)(18,33)(19,20)(22,35)(23,24)(25,34)(26,27)(31,32)(36,40)(37,38);(2,7)(3,8)(9,12)(10,18)(11,22)(13,27)(14,28)(15,26)(16,29)(17,33)(19,34)(20,25)(21,35)(23,37)(24,38)(30,39)(31,36)(32,40);(1,4,5,6)(2,3,7,8)(10,19)(16,30)(17,20)(18,34)(25,33)(29,39)(31,32)(36,40)

Polycyclic presentation:

2^9|a^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411158);