#6433320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432843
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

1712 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340516);