#8422222

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

5049 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8329418);