#9040975

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

5592 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=g,e^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, 8948171);