#6435299

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432833
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1778 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18)(19,23)(20,27)(21,30)(22,25)(24,26)(28,32)(29,31);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,21)(18,23,25,30)(20,28,29,26)(24,27,32,31);(10,13)(14,16)(17,20,22,29)(18,24,25,32)(19,26,21,28)(23,31,30,27);(9,11,12,15)(10,14,13,16)(18,25)(23,30)(24,32)(27,31);(1,3,4,7)(2,6,5,8)(18,25)(23,30)(24,32)(27,31)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,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, 6342495);