#9027848

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

5092 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)(3,7)(4,5)(6,8)(18,24)(19,29)(20,22)(27,32);(3,8)(6,7)(9,10,13,12)(11,15,16,14)(17,18)(19,30)(20,26)(21,32)(22,28)(23,24)(25,29)(27,31);(1,3,5,8)(2,6,4,7)(9,11,13,16)(10,14,12,15)(17,19)(18,25)(20,21)(22,31)(23,29)(24,30)(26,27)(28,32);(3,8)(6,7)(11,16)(14,15)(17,20,23,22)(18,26,24,28)(19,21,29,31)(25,27,30,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,21,23,31)(18,27,24,32)(19,20,29,22)(25,26,30,28);(1,4,5,2)(3,6,8,7)(9,12,13,10)(11,14,16,15)(17,22,23,20)(18,28,24,26)(19,31,29,21)(25,32,30,27)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=g,e^2=g,f^2=i,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, 8935044);