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 | C2×#3273 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2177 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)(7,11)(8,9)(10,12)(13,14)(15,22)(16,25)(17,23)(18,20)(19,27)(21,24)(26,28);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15,17,25,18,26,24,19)(14,16,21,22,20,27,23,28);(1,2,4,3)(6,9)(10,11)(13,16,18,27)(14,19,20,25)(15,23,26,21)(17,22,24,28);(1,3,4,2)(5,7,8,12)(6,10,9,11)(14,20)(16,27)(21,23)(22,28);(13,15,24,19,18,26,17,25)(14,16,23,28,20,27,21,22)Polycyclic presentation:
2^9|b^2=f,c^2=hi,d^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504111);