Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| 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 | #23438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1258 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)(6,12,15,16)(7,17)(8,21,23,24)(13,27,29,18)(19,33,31,34)(20,25)(28,36)(32,35);(5,6)(7,18)(8,22)(9,16)(10,15)(11,12)(13,20)(14,21)(17,27)(19,28)(23,26)(24,30)(25,29)(31,36)(32,34)(33,35);(2,4)(5,7,9,20,10,17,11,25)(6,13,16,27,15,29,12,18)(8,19,24,33,23,31,21,34)(14,28,22,35,30,36,26,32);(2,4)(5,8)(6,14)(7,19)(9,24)(10,23)(11,21)(12,26)(13,28)(15,30)(16,22)(17,31)(18,32)(20,33)(25,34)(27,35)(29,36)Polycyclic presentation:
2^9|a^2=h,c^2=eh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295544);