Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #388 |
| 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)(7,16,19,12)(8,20)(9,24)(10,15)(11,14)(13,22)(17,34,36,30)(18,28,25,29)(21,27,35,33)(23,32)(26,31);(1,2,3,4)(5,7)(6,12)(8,21)(9,18)(10,25)(11,19)(13,30)(14,16)(15,28)(17,22)(20,33)(23,35)(24,29)(26,36)(27,32)(31,34);(5,8)(6,13,14,31)(7,17)(9,22)(10,26)(11,23)(12,27,16,33)(15,20,24,32)(18,35)(19,36)(21,25)(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,c^b=cf,f^b=fi,g^b=gi,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72265350, #72265367, #72265386, … |
| maximal normal subgroup of | #72265350, #72265367, #72265386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31028);