Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1408 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)(5,6,12,16,11,17,10,27)(7,18,21,29,20,15,9,13)(8,22,26,32,25,14,24,31)(19,30,35,28,34,33,23,36);(1,2)(3,4)(5,7,10,9,11,20,12,21)(6,13,16,18,17,29,27,15)(8,23,26,19,25,35,24,34)(14,30,32,36,22,33,31,28);(2,4)(5,8)(6,14)(7,19)(9,23)(10,24)(11,25)(12,26)(13,28)(15,30)(16,31)(17,22)(18,33)(20,34)(21,35)(27,32)(29,36);(5,9,10,20,11,21,12,7)(6,15,16,13,17,18,27,29)(8,23,24,34,25,35,26,19)(14,30,31,28,22,33,32,36)Polycyclic presentation:
2^9|a^2=h,b^2=efgh,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,d^b=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274267);