#9706275

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

4943 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9613471);