#8599374

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4862 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8506570);