Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15172 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,27)(21,22)(24,29)(25,32)(26,30)(28,31);(9,10,12,14)(11,13,16,15)(18,22)(23,30)(24,27)(31,32);(1,2,4,3)(5,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,26)(18,23,22,30)(20,28,29,25)(24,32,27,31);(5,6,8,7)(17,20)(18,24)(19,25)(21,29)(22,27)(23,31)(26,28)(30,32);(1,3,4,2)(5,7,8,6)(9,11,12,16)(10,13,14,15)(17,18)(19,23)(20,24)(21,22)(25,31)(26,30)(27,29)(28,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fgh,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1101529);