#9040182

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

4893 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)(18,24)(19,20)(22,32)(27,30);(1,3)(2,5)(4,7)(6,8)(11,15)(13,16)(17,18,23,24)(19,25,20,29)(21,22,26,32)(27,28,30,31);(2,6)(3,7)(9,10)(11,13)(12,14)(15,16)(17,19,23,20)(18,25,24,29)(21,30,26,27)(22,31,32,28);(3,7)(5,8)(17,20,23,19)(18,25,24,29)(21,27,26,30)(22,31,32,28);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,21,23,26)(18,22,24,32)(19,27,20,30)(25,28,29,31);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,22)(18,26)(19,28)(20,31)(21,24)(23,32)(25,30)(27,29)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^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, 8947378);