#9064705

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

4738 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)(11,13)(12,16)(14,15)(17,18,23,27)(19,24,29,21)(20,28,32,25)(22,26,31,30);(1,2)(3,4)(5,6)(7,8)(10,15)(11,16)(17,19,23,29)(18,24,27,21)(20,30,32,26)(22,25,31,28);(2,6)(3,7)(9,11,14,16)(10,12,15,13)(17,20,23,32)(18,25,27,28)(19,26,29,30)(21,31,24,22);(3,7)(4,8)(9,12,14,13)(10,11,15,16)(17,21)(18,19)(20,31)(22,32)(23,24)(25,26)(27,29)(28,30);(1,3)(2,4)(5,7)(6,8)(9,10)(11,12)(13,16)(14,15)(17,22,23,31)(18,26,27,30)(19,28,29,25)(20,24,32,21);(1,4)(2,3)(5,8)(6,7)(9,13,14,12)(10,16,15,11)(17,18)(19,24)(20,25)(21,29)(22,26)(23,27)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=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, 8971901);