#9062194

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

4909 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)(17,18,23,27)(19,28,30,24)(20,31,29,25)(21,26,32,22);(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,19,23,30)(18,24,27,28)(20,22,29,26)(21,31,32,25);(1,2)(3,6)(4,5)(7,8)(10,13)(11,16)(20,29)(21,32)(22,26)(25,31);(9,11,12,16)(10,14,13,15)(17,20,23,29)(18,25,27,31)(19,26,30,22)(21,24,32,28);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,21)(18,26)(19,25)(20,28)(22,27)(23,32)(24,29)(30,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,22,23,26)(18,21,27,32)(19,29,30,20)(24,31,28,25)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=gi,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, 8969390);