#9062239

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

4917 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8969435);