#7601639

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1348 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7508835);