#6463872

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 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)(4,8)(5,7)(17,18,22,25,23,26,21,31)(19,27,29,20,28,30,32,24);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,19,23,28)(18,24,26,20)(21,29,22,32)(25,30,31,27);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(17,20,23,24)(18,19,26,28)(21,27,22,30)(25,29,31,32);(9,11,12,16)(10,14,13,15)(19,28)(20,24)(27,30)(29,32);(17,21,23,22)(18,25,26,31)(19,29,28,32)(20,27,24,30)

Polycyclic presentation:

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

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, 6371068);