#7615388

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=gi,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,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, 7522584);