#9062181

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

4888 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)(19,24)(20,30)(21,26)(22,27)(23,29)(25,31)(28,32);(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,19,22,29)(18,23,27,24)(20,21,31,28)(25,26,30,32);(1,2)(3,6)(4,5)(7,8)(10,13)(11,16)(17,19,22,29)(18,24,27,23)(20,21,31,28)(25,32,30,26);(9,11,12,16)(10,14,13,15)(17,20,22,31)(18,25,27,30)(19,28,29,21)(23,32,24,26);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,21,22,28)(18,26,27,32)(19,20,29,31)(23,25,24,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,25)(21,26)(22,27)(24,29)(28,32)(30,31)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=gi,e^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, 8969377);