#6435622

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,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, 6342818);