#6485241

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469971
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1458 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,27)(19,28,31,24)(20,25,21,26)(23,29,32,30);(9,10)(11,16)(12,15)(13,14)(17,19,22,31)(18,23,27,32)(20,30,21,29)(24,26,28,25);(2,5)(6,8)(10,14)(15,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,25)(26,32)(27,28);(9,11,13,12)(10,15,14,16)(17,20,22,21)(18,25,27,26)(19,29,31,30)(23,28,32,24);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(17,20,22,21)(18,26,27,25)(19,30,31,29)(23,28,32,24)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6392437);