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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2307 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)(15,22)(17,25)(23,26)(24,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,23)(16,20)(17,24)(18,21)(19,27)(22,26)(25,28);(1,2,3,4)(13,15,18,22)(14,17,21,25)(16,24,19,28)(20,26,27,23);(6,10)(9,11)(13,16,18,19)(14,20,21,27)(15,24,22,28)(17,26,25,23);(7,12)(9,11)(13,17,18,25)(14,15,21,22)(16,23,19,26)(20,28,27,24)Polycyclic presentation:
2^9|c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,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, 6589492);