#6536852

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1486 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)(17,18,22,25)(19,26,29,32)(20,24,21,31)(23,30,27,28);(9,10)(11,14)(12,13)(15,16)(17,19,20,30,22,29,21,28)(18,23,31,32,25,27,24,26);(2,6)(7,8)(10,13)(14,16)(17,18)(19,27)(20,31)(21,24)(22,25)(23,29)(26,30)(28,32);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,21)(18,24,25,31)(19,28,29,30)(23,32,27,26);(1,4,5,3)(2,8,6,7)(11,15)(14,16)

Polycyclic presentation:

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

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, 6444048);