#8599558

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

4973 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)(11,15)(12,16)(13,14)(17,18)(19,21)(20,30)(22,25)(23,26)(24,28)(27,29)(31,32);(2,5)(6,8)(9,11,13,16)(10,12,14,15)(17,19,23,24)(18,21,26,28)(20,31,25,27)(22,29,30,32);(1,2,4,5)(3,6,7,8)(9,12,13,15)(10,11,14,16)(17,20,23,25)(18,22,26,30)(19,27,24,31)(21,32,28,29);(3,7)(6,8)(9,10,13,14)(11,15,16,12)(17,18)(19,28)(20,22)(21,24)(23,26)(25,30)(27,29)(31,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,21,23,28)(18,24,26,19)(20,29,25,32)(22,27,30,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,14)(15,16)(17,22)(18,25)(19,29)(20,26)(21,27)(23,30)(24,32)(28,31)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=g,e^2=gh,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, 8506754);