#7593020

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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1656 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=g,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, 7500216);