#7560728

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
prime radical	#61178	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1740 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)(3,6)(4,5)(7,8)(9,10)(11,20)(12,24)(13,27)(14,32)(15,19)(16,23)(17,30)(18,25)(21,37)(22,34)(26,35)(28,38)(29,33)(31,40)(36,39);(2,5)(6,8)(9,11)(10,16)(12,25)(13,28)(14,22)(15,23)(17,33)(18,34)(19,20)(21,31)(24,32)(26,30)(27,37)(29,39)(35,36)(38,40);(1,3,4,7)(2,6,5,8)(9,12,25,11,15,22,14,23)(10,16,32,34,19,20,18,24)(13,29,39,28,31,26,30,21)(17,35,40,38,36,33,27,37);(9,13,14,30,15,31,25,39)(10,17,18,27,19,36,32,40)(11,21,12,26,23,28,22,29)(16,33,24,38,20,35,34,37);(2,5)(6,8)(9,14,15,25)(10,18,19,32)(11,22,23,12)(13,30,31,39)(16,34,20,24)(17,27,36,40)(21,29,28,26)(33,37,35,38)

Polycyclic presentation:

2^9|c^2=fg,d^2=g,e^2=i,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,d^c=di,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, 7467924);