#6573309

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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)(17,18,23,32,22,26,21,25)(19,27,28,20,24,30,29,31);(9,11,15,13)(10,16,14,12)(17,19,22,24)(18,20,26,31)(21,28,23,29)(25,30,32,27);(2,6)(7,8)(9,12,15,16)(10,11,14,13)(17,20)(18,24)(19,26)(21,27)(22,31)(23,30)(25,28)(29,32);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,16,14,12)(17,21,22,23)(18,25,26,32)(19,28,24,29)(20,30,31,27);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,12,13,16)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,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^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, 6480505);