#6446471

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1470 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)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,25)(21,30)(22,26)(23,27)(28,32)(29,31);(2,5)(6,8)(19,20)(21,22)(24,25)(26,30)(28,29)(31,32);(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,21)(22,29)(23,28)(25,30)(26,31)(27,32);(17,20,23,29)(18,25,27,31)(19,21,28,22)(24,30,32,26);(1,3,4,7)(2,6,5,8)(17,21,23,22)(18,26,27,30)(19,20,28,29)(24,31,32,25)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6353667);