Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1440 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,22,26,32)(9,15)(10,14)(11,16)(12,19)(13,25,31,24)(18,27,23,30)(20,28)(21,29)(33,34,36,35);(5,7,10,19,11,21,9,20)(6,12,14,29,16,28,15,17)(8,23,24,34,26,18,25,35)(13,30,32,36,31,27,22,33);(2,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,22)(15,32)(16,31)(17,33)(19,34)(20,35)(21,23)(28,30)(29,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=di,f^b=fi,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237180, #72237182, #72237192, … |
| maximal normal subgroup of | #72237180, #72237182, #72237192, … |
| maximal quotient of | #72239093 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30818);