#9028241

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

4900 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)(11,14)(12,13)(15,16)(17,18,22,26)(19,20,24,23)(21,25,29,32)(27,28,31,30);(1,2,5,4)(3,7,8,6)(9,11,13,15)(10,14,12,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32);(1,3,5,8)(2,6,4,7)(10,12)(11,15)(19,24)(20,23)(21,29)(25,32);(3,8)(6,7)(9,11,13,15)(10,14,12,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,30)(28,32)(29,31);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,23)(18,24,26,19)(21,28,29,30)(25,31,32,27);(1,4,5,2)(3,6,8,7)(9,12,13,10)(11,16,15,14)(17,21,22,29)(18,25,26,32)(19,27,24,31)(20,28,23,30)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8935437);