Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,12)(9,10)(13,14)(15,17)(16,21)(18,20)(19,24)(22,25)(23,27)(26,28);(2,3)(6,10)(11,12)(14,15)(16,23)(18,25)(19,27)(21,28)(24,26);(1,2,4,3)(5,7,9,8)(6,11,10,12)(13,15,17,14)(16,21,24,19)(18,22,25,20)(23,27,26,28);(1,3,4,2)(5,7,9,8)(6,11,10,12)(13,16)(14,19)(15,21)(17,24)(18,26)(20,27)(22,28)(23,25);(5,8,9,7)(6,11,10,12)(14,15)(19,21)(20,22)(27,28)Polycyclic presentation:
2^9|c^2=fgi,d^2=fg,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472499);