#7610197

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	#45094	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1535 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)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,22)(19,24,29,30)(20,25,28,31)(21,27,32,26);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,29)(18,24,22,30)(20,27,28,26)(21,25,32,31);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20)(18,25)(19,27)(21,30)(22,31)(23,28)(24,32)(26,29);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,21,22,27,23,32,18,26)(19,28,24,25,29,20,30,31);(2,6)(7,8)(20,28)(21,32)(25,31)(26,27)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=fh,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, 7517393);