#9040240

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

4967 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,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, 8947436);