Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#3002 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3761 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(10,14,16,19)(11,17,18,20);(1,2,3,4)(9,10,12,19,13,16,21,14)(11,15,20,24,18,23,17,22);(9,11)(10,15)(12,20)(13,18)(14,22)(16,23)(17,21)(19,24);(6,8)(10,16)(14,19)(15,23)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=eh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,d^b=di,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286571 |
| maximal normal subgroup of | #72286571 |
| maximal quotient of | #72286602, #72286630 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299824);