#6435654

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432838
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

1596 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)(17,18,22,25,23,26,21,24)(19,20,27,30,29,31,28,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,28)(22,27)(23,29)(24,32)(25,30)(26,31);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,19)(21,30)(22,32)(23,31)(24,27)(25,28)(26,29);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,24,26,25)(19,27,29,28)(20,30,31,32);(17,21,23,22)(18,25,26,24)(19,28,29,27)(20,30,31,32)

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,d^b=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6342850);