#9040720

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

4843 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=h,e^2=h,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, 8947916);