#6450310

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1510 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)(3,8)(4,6)(5,7)(9,10,12,13)(11,14,15,16)(17,18)(19,22)(20,25)(21,27)(23,26)(24,30)(28,29)(31,32);(1,3,5,4)(2,6,7,8)(17,19,22,18)(20,28,30,32)(21,27,23,26)(24,29,25,31);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20,22,30)(18,24,19,25)(21,29,23,31)(26,28,27,32);(9,11,12,15)(10,14,13,16)(17,20,22,30)(18,25,19,24)(21,29,23,31)(26,32,27,28);(17,21,22,23)(18,26,19,27)(20,29,30,31)(24,28,25,32)

Polycyclic presentation:

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

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, 6357506);