#9040222

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

4971 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,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, 8947418);