#6573272

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1408 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)(3,7)(4,8)(5,6)(9,10,15,14)(11,12,13,16)(17,18,24,30,23,19,22,27)(20,25,32,29,31,28,21,26);(9,11,15,13)(10,16,14,12)(17,19,23,18)(20,25,31,28)(21,29,32,26)(22,27,24,30);(2,6)(7,8)(9,12,15,16)(10,11,14,13)(17,20,23,31)(18,25,19,28)(21,24,32,22)(26,30,29,27);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,16,14,12)(17,21,23,32)(18,26,19,29)(20,22,31,24)(25,27,28,30);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,12,13,16)(20,31)(21,32)(25,28)(26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480468);