#6535278

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 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)(10,13)(14,16)(17,18,23,19)(20,28,30,32)(21,25,22,26)(24,31,27,29);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,18)(20,27,30,24)(21,25,22,26)(28,29,32,31);(10,13)(14,16)(17,20,23,30)(18,24,19,27)(21,29,22,31)(25,32,26,28);(1,3,4,7)(2,6,5,8)(11,15)(14,16)(17,21,23,22)(18,25,19,26)(20,29,30,31)(24,32,27,28);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,26,19,25)(20,29,30,31)(24,28,27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6442474);