#7616422

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1197 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=gi,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

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, 7523618);