#7516562

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1923 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)(11,16)(12,23)(13,26)(14,28)(15,18)(17,32)(19,20)(21,37)(22,36)(24,33)(25,31)(27,30)(29,35)(34,40)(38,39);(2,5)(6,8)(9,11,15,22)(10,16,18,36)(12,24,25,34)(13,20,27,39)(14,29,32,21)(17,37,28,35)(19,30,38,26)(23,33,31,40);(9,12,15,25)(10,13,18,27)(11,19,22,38)(14,30,32,26)(16,33,36,40)(17,31,28,23)(20,35,39,37)(21,34,29,24);(1,3,4,7)(2,6,5,8)(9,13,15,27)(10,12,18,25)(11,20,22,39)(14,31,32,23)(16,34,36,24)(17,30,28,26)(19,35,38,37)(21,33,29,40);(2,5)(6,8)(12,25)(13,27)(19,38)(20,39)(23,31)(24,34)(26,30)(33,40)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,e^c=ei,g^c=gi,h^c=hi,e^d=ei,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, 7423758);