Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184811 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C27 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C27 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
9898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,19,20,17)(16,18);(1,2)(3,4)(6,9)(10,12);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,17)(16,20)(18,19);(6,9)(10,12)Polycyclic presentation:
2^9|a^2=h,c^2=e,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72217617, #72217666, #72222882, … |
| maximal normal subgroup of | #72217617, #72217666, #72222882, … |
| maximal quotient of | #72222876, #72255630, #72264333, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266995);