#9040168

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

4908 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^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, 8947364);