Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×C22⋊C4 |
| commutator | C2×C4 | C43 |
| Frattini, cosocle | C22×C42 | C23 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,4)(10,14)(11,12)(15,16)(17,18)(19,23)(20,22)(21,24);(9,10,12,16,13,15,11,14)(18,22)(19,20)(23,24);(5,6,7,8)(10,15)(14,16)(17,19,21,20)(18,23,24,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=h,f^2=hi,b^a=bd,d^a=dh,f^a=fhi,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72217677, #72279409, #72279503, … |
| maximal normal subgroup of | #72217677, #72279409, #72279503, … |
| maximal quotient of | #72213034 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12645);