Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#3475 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3554 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)(5,6)(7,10)(8,9)(11,12)(13,14,19,23,18,15,20,26)(16,24,28,25,27,21,17,22);(1,2,3,4)(13,15,19,26,18,14,20,23)(16,21,28,22,27,24,17,25);(6,9)(10,12)(13,16)(14,21)(15,24)(17,20)(18,27)(19,28)(22,23)(25,26);(5,7,8,11)(6,10,9,12)(13,17,18,28)(14,22,15,25)(16,20,27,19)(21,23,24,26);(16,27)(17,28)(21,24)(22,25)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ci,e^c=ei,e^d=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279562);