#6446893

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1717 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)(11,14)(12,16)(13,15)(17,18,21,25)(19,20,24,23)(22,26,31,32)(27,28,30,29);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,18)(19,27)(20,29)(21,25)(22,32)(23,28)(24,30)(26,31);(9,12,13,11)(10,14,15,16)(17,19,21,24)(18,23,25,20)(22,29,31,28)(26,30,32,27);(1,3,4,7)(2,6,5,8)(17,20,21,23)(18,24,25,19)(22,27,31,30)(26,28,32,29);(18,25)(19,24)(26,32)(28,29)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,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, 6354089);