#6633377

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632036
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×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1846 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)(3,5)(4,6)(7,8)(9,10,11,13)(12,14,16,15)(17,18,22,25)(19,26,28,23)(20,24,21,29)(27,32,30,31);(1,3)(2,5)(4,7)(6,8)(10,13)(14,15)(17,19,22,28)(18,23,25,26)(20,27,21,30)(24,31,29,32);(3,7)(5,8)(10,14)(12,16)(13,15)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,26,32);(3,7)(5,8)(10,13)(14,15)(17,21,22,20)(18,24,25,29)(19,27,28,30)(23,32,26,31);(2,6)(5,8)(18,25)(23,26)(24,29)(31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fi,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6540573);