Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1440 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6)(7,17,21,12)(8,22)(9,16)(10,15)(11,14)(13,24)(18,34,35,30)(19,28,20,29)(23,33,36,27)(25,32)(26,31);(1,2,3,4)(5,7)(6,12)(8,23)(9,20)(10,19)(11,21)(13,30)(14,17)(15,29)(16,28)(18,24)(22,27)(25,36)(26,35)(31,34)(32,33);(5,8)(6,13,14,31)(7,18)(9,24)(10,26)(11,25)(12,27,17,33)(15,22,16,32)(19,23)(20,36)(21,35)(28,30,29,34)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=d,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31044);