Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60422 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2626 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,10,15,11,8,20,14)(7,12,19,26,18,16,30,25)(9,21,22,28,23,13,27,29)(17,31,33,35,32,24,34,36);(1,2,3,4)(5,7)(6,12)(8,16)(9,17)(10,19)(11,18)(13,24)(14,25)(15,26)(20,30)(21,31)(22,33)(23,32)(27,34)(28,35)(29,36);(5,8)(6,11)(7,16)(9,21)(10,14)(12,18)(13,23)(15,20)(17,31)(19,25)(22,28)(24,32)(26,30)(27,29)(33,35)(34,36);(5,9,10,22,11,23,20,27)(6,13,15,29,8,21,14,28)(7,17,19,33,18,32,30,34)(12,24,26,36,16,31,25,35);(7,18)(12,16)(17,32)(19,30)(24,31)(25,26)(33,34)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,e^b=ei,g^c=gi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72250686, #72252972, #72256876, … |
| maximal normal subgroup of | #72250686, #72252972, #72256876, … |
| maximal quotient of | #72318352, #72320091, C2×#6372840, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280036);