#7597048

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563337
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

1987 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)(3,7)(4,5)(6,8)(9,10)(11,21)(12,26)(13,17)(14,31)(15,20)(16,23)(18,29)(19,28)(22,32)(24,35)(25,34)(27,38)(30,37)(33,40)(36,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,23,15,25,28,24)(10,16,19,21,20,35,31,34)(12,27,13,22,29,39,30,40)(17,36,26,32,37,38,18,33);(2,5)(6,7)(9,12,14,13,15,29,28,30)(10,17,19,26,20,37,31,18)(11,22,24,27,25,40,23,39)(16,32,34,36,35,33,21,38);(1,3,4,8)(2,6,5,7)(9,12,28,30,15,29,14,13)(10,17,31,18,20,37,19,26)(11,22,23,39,25,40,24,27)(16,32,21,38,35,33,34,36);(9,13,14,29,15,30,28,12)(10,18,19,17,20,26,31,37)(11,22,23,39,25,40,24,27)(16,33,21,36,35,32,34,38)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=gi,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,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, 7504244);