Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
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×#3521 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,9)(10,12)(13,14,17,21)(15,22,24,27)(16,20,18,25)(19,26,28,23);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,25)(17,21)(18,20)(22,23)(24,28)(26,27);(13,15)(14,19)(16,23)(17,24)(18,26)(20,27)(21,28)(22,25);(6,9)(10,12)(13,16,17,18)(14,20,21,25)(15,23,24,26)(19,27,28,22);(5,7,8,11)(6,10,9,12)(14,21)(19,28)(20,25)(22,27)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,d^b=df,e^b=ei,g^c=gi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286791);