Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| 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×#424 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6,10,17,11,15,27,16)(7,12,20,30,21,28,35,29)(8,22,25,9,14,26,24,13)(18,32,34,31,23,36,33,19);(2,3)(5,7,10,20,11,21,27,35)(6,12,17,30,15,28,16,29)(8,23,25,33,14,18,24,34)(9,19,26,32,13,31,22,36);(5,8)(6,13)(7,18)(9,15)(10,25)(11,14)(12,19)(16,26)(17,22)(20,34)(21,23)(24,27)(28,31)(29,36)(30,32)(33,35);(1,3,4,2)(5,9)(6,14)(7,19)(8,15)(10,26)(11,13)(12,18)(16,25)(17,24)(20,32)(21,31)(22,27)(23,28)(29,33)(30,34)(35,36);(6,15)(9,13)(12,28)(16,17)(19,31)(22,26)(29,30)(32,36)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ci,d^b=df,g^c=gi,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72250686, #72252980, #72256865, … |
| maximal normal subgroup of | #72250686, #72252980, #72256865, … |
| maximal quotient of | #72318348, #72320007, #72320103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289102);