#7597041

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563333
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

1989 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)(3,7)(4,5)(6,8)(10,17)(11,21)(12,13)(14,15)(18,32)(19,35)(20,31)(23,39)(25,26)(28,29)(33,37)(34,36);(1,2)(3,6)(4,5)(7,8)(9,10,14,17,16,20,15,31)(11,22,25,32,26,38,21,18)(12,27,28,35,29,40,13,19)(23,34,24,37,39,33,30,36);(2,5)(6,7)(9,11,15,21,16,26,14,25)(10,18,17,22,20,32,31,38)(12,23,13,30,29,39,28,24)(19,33,27,36,35,34,40,37);(1,3,4,8)(2,6,5,7)(9,12,15,13,16,29,14,28)(10,19,17,27,20,35,31,40)(11,23,21,30,26,39,25,24)(18,33,22,36,32,34,38,37);(9,13,14,12,16,28,15,29)(10,19,17,27,20,35,31,40)(11,24,25,39,26,30,21,23)(18,34,22,37,32,33,38,36)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=g,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,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, 7504237);