#6433827

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433044
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1794 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:

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

Polycyclic presentation:

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

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, 6341023);