Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(6,8)(9,10,13,15)(11,12,16,14)(18,22)(23,26)(24,28)(30,32);(9,11,13,16)(10,14,15,12)(17,18,21,22)(19,23,25,26)(20,28,29,24)(27,32,31,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(19,25)(23,26)(27,31)(30,32);(3,7)(6,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,3,4,7)(2,6,5,8)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,32)(29,31);(1,2)(3,6)(4,5)(7,8)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10232603);