#9040510

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

4933 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)(4,8)(5,6)(9,10)(11,12)(13,16)(14,15)(17,18)(19,24)(20,22)(21,26)(23,25)(27,31)(28,32)(29,30);(1,2,5,6)(3,4,8,7)(9,11)(10,13)(12,14)(15,16)(17,19,23,28)(18,24,25,32)(20,30,27,21)(22,29,31,26);(3,8)(4,7)(9,12)(10,16)(11,14)(13,15)(18,25)(19,28)(21,30)(22,31);(2,6)(3,8)(9,10,15,14)(11,12,16,13)(17,20,23,27)(18,22,25,31)(19,21,28,30)(24,26,32,29);(1,3,5,8)(2,7,6,4)(9,13,15,12)(10,16,14,11)(17,21)(18,26)(19,20)(22,24)(23,30)(25,29)(27,28)(31,32);(1,4,5,7)(2,8,6,3)(9,14,15,10)(11,12,16,13)(17,22)(18,27)(19,29)(20,25)(21,24)(23,31)(26,28)(30,32)

Polycyclic presentation:

2^9|b^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,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, 8947706);