Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,11,12,10)(13,14)(15,17)(16,26)(18,21)(19,20)(22,25)(23,28)(24,27);(5,6)(7,10)(8,9)(11,12)(13,14,18,21)(15,22,24,28)(16,20,19,26)(17,23,27,25);(1,2,3,4)(6,9)(10,11)(13,15,18,24)(14,17,21,27)(16,25,19,23)(20,28,26,22);(5,7,8,12)(6,10,9,11)(13,16,18,19)(14,20,21,26)(15,23,24,25)(17,22,27,28);(13,17,18,27)(14,15,21,24)(16,22,19,28)(20,23,26,25)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591921);