Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,9)(10,12)(14,18)(15,21)(17,24)(19,26)(22,27)(23,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,22)(16,18)(17,20)(19,23)(21,26)(24,25)(27,28);(13,15,17,21,16,23,24,28)(14,19,20,27,18,22,25,26);(6,9)(10,12)(14,18)(19,22)(20,25)(26,27);(5,7,8,11)(6,10,9,12)(14,18)(19,22)(20,25)(26,27)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ci,d^b=df,e^b=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287592);