Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
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×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1147 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)(5,6,12,16,11,17,10,18)(7,19,24,31,23,33,22,13)(8,25,28,36,21,14,27,32)(9,29,30,34,26,35,20,15);(2,4)(5,7)(6,13,17,31)(8,26)(9,21)(10,22)(11,23)(12,24)(14,15,25,34)(16,19,18,33)(20,28)(27,30)(29,36,35,32);(2,4)(5,8)(6,14)(7,20)(9,24)(10,27)(11,21)(12,28)(13,29)(15,33)(16,32)(17,25)(18,36)(19,34)(22,26)(23,30)(31,35);(2,4)(5,9,12,30,11,26,10,20)(6,15,18,35,17,34,16,29)(7,21,24,27,23,8,22,28)(13,25,33,32,31,14,19,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=ef,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, 273402);