Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
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 | C23×C22⋊C4 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4700 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)(5,6,10,15,11,14,9,24)(7,12,19,27,17,28,18,34)(8,20,22,13,23,31,25,30)(16,32,33,26,21,36,35,29);(1,2)(3,4)(5,7)(6,12)(8,21)(9,18)(10,19)(11,17)(13,29)(14,28)(15,27)(16,23)(20,36)(22,35)(24,34)(25,33)(26,30)(31,32);(5,8)(6,13)(7,16)(9,25)(10,22)(11,23)(12,26)(14,30)(15,31)(17,21)(18,35)(19,33)(20,24)(27,36)(28,29)(32,34);(2,4)(5,7)(6,12)(8,16)(9,18)(10,19)(11,17)(13,26)(14,28)(15,27)(20,32)(21,23)(22,33)(24,34)(25,35)(29,30)(31,36);(6,14)(7,17)(13,30)(15,24)(16,21)(18,19)(20,31)(33,35)Polycyclic presentation:
2^9|a^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ci,d^b=df,e^b=ei,g^c=gi,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72311254, #72312225, #72318864, … |
| maximal normal subgroup of | #72311254, #72312225, #72318864, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286310);