Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| 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 | |
1086 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)(5,6,12,16,11,17,10,28)(7,15,9,27,20,34,19,13)(8,21,26,33,25,14,24,32)(18,30,23,29,35,31,22,36);(2,4)(5,7,12,9,11,20,10,19)(6,13,28,34,17,27,16,15)(8,22,24,35,25,23,26,18)(14,30,32,29,21,31,33,36);(2,4)(5,8)(6,14)(7,18)(9,23)(10,24)(11,25)(12,26)(13,29)(15,31)(16,32)(17,21)(19,22)(20,35)(27,36)(28,33)(30,34);(2,4)(5,9,12,20,11,19,10,7)(6,15,28,13,17,34,16,27)(8,23,26,35,25,22,24,18)(14,31,33,29,21,30,32,36)Polycyclic presentation:
2^9|a^2=h,b^2=egh,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,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, 275130);