Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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×#3562 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(7,11)(13,14,18,22,17,23,16,21)(15,24,25,20,26,19,27,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,27)(17,26)(18,25)(20,21)(22,28)(23,24);(1,2,3,4)(13,15)(14,20)(16,25)(17,26)(18,27)(19,21)(22,24)(23,28);(6,9)(10,12)(13,16,17,18)(14,21,23,22)(15,25,26,27)(19,28,24,20);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,22,23,21)(15,25,26,27)(19,20,24,28)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=df,e^b=ei,g^c=gi,h^c=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300952);