Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | |
2039 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,12)(9,10);(1,2)(3,4)(6,10)(11,12)(13,14)(15,20)(16,26)(17,23)(18,22)(19,24)(21,27)(25,28);(2,4)(5,7,9,8)(6,11,10,12)(13,15,19,16)(14,20,24,26)(17,27,18,25)(21,22,28,23);(2,4)(13,16,19,15)(14,21,24,28)(17,27,18,25)(20,22,26,23);(5,8,9,7)(6,11,10,12)(13,17,19,18)(14,22,24,23)(15,25,16,27)(20,21,26,28)Polycyclic presentation:
2^9|c^2=fi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7427154);