Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 | #45134 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,23)(16,25)(17,21)(18,22)(19,24)(20,27)(26,28);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,22)(17,24)(18,20)(21,23)(25,26)(27,28);(6,10)(7,11)(13,14)(15,19)(16,20)(17,21)(18,22)(23,24)(25,27)(26,28);(6,10)(9,12)(13,16,26,23,17,27,18,19)(14,20,28,24,21,25,22,15);(1,3,4,2)(14,21)(16,27)(19,23)(22,28)Polycyclic presentation:
2^9|b^2=fi,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7518743);