#7613880

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
prime radical	#61415	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 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)(4,5)(6,8)(9,10)(11,22)(12,25)(13,28)(14,21)(15,20)(16,19)(17,24)(18,31)(23,39)(26,35)(27,37)(29,40)(30,34)(32,38)(33,36);(2,5)(6,7)(9,11,15,24)(10,17,20,22)(12,16,27,14)(13,29,32,23)(18,34,36,35)(19,25,21,37)(26,33,30,31)(28,39,38,40);(9,12,16,24,15,27,14,11)(10,17,21,25,20,22,19,37)(13,30,31,23,32,26,33,29)(18,35,28,39,36,34,38,40);(9,13,15,32)(10,18)(11,23,24,29)(12,26,27,30)(14,33,16,31)(17,34)(19,38)(20,36)(21,28)(22,35)(25,40)(37,39);(1,3,4,8)(2,6,5,7)(9,14,15,16)(10,19,20,21)(11,12,24,27)(13,31,32,33)(17,25,22,37)(18,28,36,38)(23,30,29,26)(34,39,35,40)

Polycyclic presentation:

2^9|b^2=i,c^2=g,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,d^c=di,e^c=ei,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, 7521076);