Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #42951 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1974 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)(13,14,18,19)(15,23,24,20)(16,25,26,21)(17,27,28,22);(6,9)(7,11)(13,15,18,24)(14,20,19,23)(16,22,26,27)(17,25,28,21);(1,2,3,4)(13,16,18,26)(14,21,19,25)(15,17,24,28)(20,27,23,22);(6,9)(10,12)(15,24)(16,26)(20,23)(21,25);(5,7,8,11)(6,10,9,12)(13,17,18,28)(14,22,19,27)(15,16,24,26)(20,25,23,21)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592387);