#7560688

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519063
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	#61169	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1737 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)(11,16)(12,25)(13,23)(14,18)(15,20)(17,28)(19,31)(21,34)(22,30)(24,33)(26,32)(27,39)(29,35)(36,37)(38,40);(2,5)(6,8)(9,11,15,24)(10,16,20,33)(12,26,29,38)(13,18,30,19)(14,22,31,23)(17,34,36,27)(21,37,39,28)(25,32,35,40);(1,3,4,7)(2,6,5,8)(9,12,14,28,15,29,31,37)(10,17,19,25,20,36,18,35)(11,21,23,26,24,39,22,38)(13,27,16,32,30,34,33,40);(9,13,14,16,15,30,31,33)(10,11,19,23,20,24,18,22)(12,27,28,32,29,34,37,40)(17,21,25,26,36,39,35,38);(2,5)(6,8)(9,14,15,31)(10,18,20,19)(11,22,24,23)(12,28,29,37)(13,16,30,33)(17,35,36,25)(21,38,39,26)(27,32,34,40)

Polycyclic presentation:

2^9|b^2=i,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,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7467884);