#7590921

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1304 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,c^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,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, 7498117);