#8657237

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

5009 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=hi,d^2=gi,e^2=gi,f^2=hi,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, 8564433);