#7564818

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

1800 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)(9,10,15,19)(11,21,25,17)(12,26,30,28)(13,14,20,16)(18,32,36,29)(22,38,40,35)(23,24,31,33)(27,37,39,34);(2,6)(7,8)(9,11,15,25)(10,17,19,21)(12,27,30,39)(13,31,20,23)(14,24,16,33)(18,35,36,38)(22,32,40,29)(26,34,28,37);(1,3,5,4)(2,7,6,8)(9,12,14,32,15,30,16,29)(10,18,13,28,19,36,20,26)(11,22,33,39,25,40,24,27)(17,34,23,38,21,37,31,35);(9,13,16,10,15,20,14,19)(11,23,24,17,25,31,33,21)(12,28,29,18,30,26,32,36)(22,38,27,34,40,35,39,37);(1,4,5,3)(2,7,6,8)(9,14,15,16)(10,13,19,20)(11,24,25,33)(12,29,30,32)(17,31,21,23)(18,26,36,28)(22,39,40,27)(34,38,37,35)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=g,e^2=f,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, 7472014);