#7564811

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519083
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

1835 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=g,e^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,h^d=hi

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, 7472007);