#8720326

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

5151 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627522);