Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | #42474 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2362 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)(7,11)(8,10)(9,12)(13,14)(15,20)(16,25)(17,22)(18,21)(19,23)(24,27)(26,28);(5,7)(6,9)(8,12)(10,11)(15,24)(16,18)(17,26)(20,27)(21,25)(22,28);(1,2,3,4)(13,15,19,22)(14,20,23,17)(16,24,18,28)(21,26,25,27);(6,10)(9,11)(13,16,19,18)(14,21,23,25)(15,24,22,28)(17,27,20,26);(7,12)(9,11)(13,17,19,20)(14,22,23,15)(16,26,18,27)(21,24,25,28)Polycyclic presentation:
2^9|c^2=gi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589967);