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×#3355 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1799 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)(5,6,9,8)(7,10,12,11)(14,19)(15,23)(16,17)(20,28)(21,22)(24,27);(2,3)(5,7)(6,10)(8,11)(9,12)(13,14,17,19,18,22,16,21)(15,24,25,28,23,20,26,27);(1,2,4,3)(5,8,9,6)(7,11,12,10)(13,15)(14,20)(16,26)(17,25)(18,23)(19,27)(21,28)(22,24);(1,3,4,2)(7,12)(10,11)(13,16,18,17)(14,21,22,19)(15,25,23,26)(20,27,24,28);(13,14,16,21,18,22,17,19)(15,20,26,28,23,24,25,27)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=h,d^2=g,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, 7511542);