#6435650

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432833
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 ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1762 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,21,25,22,24,20,29)(19,26,28,31,27,32,30,23);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(19,27)(23,31)(26,32)(28,30);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,19)(18,23)(20,28)(21,30)(22,27)(24,31)(25,32)(26,29);(9,11,12,15)(10,14,13,16)(18,24)(23,31)(25,29)(26,32);(17,20,22,21)(18,25,24,29)(19,28,27,30)(23,32,31,26)

Polycyclic presentation:

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

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, 6342846);