#6432674

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431843
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1703 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339870);