#9063243

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

4877 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8970439);