#6579842

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#36009	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

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, 6487038);