#7614262

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563327
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	#3496	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1729 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)(4,8)(5,6)(9,10)(11,23)(12,27)(13,31)(14,22)(15,21)(16,20)(17,26)(18,30)(19,28)(24,40)(25,37)(29,36)(32,38)(33,35)(34,39);(2,6)(7,8)(9,11,15,26)(10,17,21,23)(12,28,29,33)(13,24,32,30)(14,25,16,34)(18,38,40,31)(19,27,35,36)(20,37,22,39);(1,3,5,4)(2,7,6,8)(9,12,16,30,15,29,14,24)(10,18,22,27,21,40,20,36)(11,13,25,28,26,32,34,33)(17,35,37,38,23,19,39,31);(9,13)(10,19,21,35)(11,24)(12,25)(14,33)(15,32)(16,28)(17,36,23,27)(18,39,40,37)(20,38,22,31)(26,30)(29,34);(1,4,5,3)(2,7,6,8)(9,14,15,16)(10,20,21,22)(11,25,26,34)(12,24,29,30)(13,28,32,33)(17,37,23,39)(18,36,40,27)(19,31,35,38)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=h,e^2=f,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,e^d=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, 7521458);