Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(7,11)(13,14,19,23)(15,20,22,17)(16,21,27,18)(24,26,28,25);(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,21)(18,27)(19,22)(20,23)(24,26)(25,28);(1,2,3,4)(6,9)(10,12)(13,16,17,25,19,27,20,26)(14,21,15,24,23,18,22,28);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,22,23,15)(16,26,27,25)(18,28,21,24);(13,18,17,28,19,21,20,24)(14,16,15,25,23,27,22,26)Polycyclic presentation:
2^9|a^2=i,c^2=fgh,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724295);