Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184979 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×2−1+4 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
15810 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)(7,15)(8,19,21,12)(9,22)(10,14)(11,17)(13,24)(16,30,32,29)(18,28)(20,27,36,34)(23,26)(25,31,33,35);(5,7,10,18)(6,11,14,26)(8,16,21,32)(9,23,24,17)(12,25,19,33)(13,28,22,15)(20,35,36,31)(27,30,34,29);(5,8)(6,12)(7,16)(9,20)(10,21)(11,25)(13,27)(14,19)(15,29)(17,31)(18,32)(22,34)(23,35)(24,36)(26,33)(28,30);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72257736, #72257744, #72263944, … |
| maximal normal subgroup of | #72257736, #72257744, #72263944, … |
| maximal quotient of | #72264358, #72302879, #72302907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299329);