#9041005

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

5307 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)(3,6)(4,5)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18,23,27)(19,24,28,22)(20,25,29,32)(21,26,30,31);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19,23,28)(18,24,27,22)(20,30,29,21)(25,31,32,26);(2,4)(3,7)(9,12,13,14)(10,11,16,15)(19,28)(21,30)(22,24)(26,31);(3,7)(6,8)(9,12,13,14)(10,15,16,11)(17,20)(18,25)(19,21)(22,31)(23,29)(24,26)(27,32)(28,30);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,14)(15,16)(17,21)(18,26)(19,29)(20,28)(22,25)(23,30)(24,32)(27,31);(1,4,5,2)(3,8,7,6)(9,11)(10,12)(13,15)(14,16)(17,22)(18,19)(20,31)(21,25)(23,24)(26,29)(27,28)(30,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=g,d^2=g,f^2=h,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, 8948201);