Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(10,12)(13,14,18,23,17,22,25,28)(15,24,16,21,26,19,27,20);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,18)(17,26)(20,23)(21,28)(22,24)(25,27);(1,2,3,4)(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,25)(17,27)(18,26)(19,28)(21,22)(23,24);(6,9)(10,12)(13,16)(14,21)(15,25)(17,27)(18,26)(19,23)(20,22)(24,28);(15,26)(16,27)(19,24)(20,21)Polycyclic presentation:
2^9|a^2=h,c^2=fg,h^2=i,b^a=bf,c^a=cg,c^b=ci,d^b=df,e^b=ei,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287407);