#7613478

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563341
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

1913 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|c^2=fg,d^2=hi,e^2=i,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

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, 7520674);