Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,12,16,11,17,10,30)(7,18,22,31,21,32,20,13)(8,23,26,35,25,14,24,34)(9,27,29,33,28,36,19,15);(5,7)(6,13,17,31)(8,9)(10,20)(11,21)(12,22)(14,33,23,15)(16,18,30,32)(19,24)(25,28)(26,29)(27,34,36,35);(2,4)(5,8)(6,14)(7,19)(9,22)(10,24)(11,25)(12,26)(13,27)(15,32)(16,34)(17,23)(18,33)(20,28)(21,29)(30,35)(31,36);(5,9,10,19,11,28,12,29)(6,15,16,27,17,33,30,36)(7,8,20,24,21,25,22,26)(13,14,18,34,31,23,32,35)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272220);