#6630861

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629007
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1782 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)(9,10,12,14)(11,15,16,13)(17,18,22,25)(19,26,29,32)(20,24,30,31)(21,28,23,27);(2,5)(6,8)(17,19,23,30,22,29,21,20)(18,24,28,32,25,31,27,26);(9,11,12,16)(10,13,14,15)(18,25)(24,31)(26,32)(27,28);(1,3,4,7)(2,6,5,8)(17,20,21,29,22,30,23,19)(18,26,27,31,25,32,28,24);(19,29)(20,30)(24,31)(26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,e^b=ei,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538057);