Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,11)(13,14,17,21)(15,20,23,18)(16,24,27,28)(19,25,26,22);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15,17,23)(14,18,21,20)(16,25,27,22)(19,28,26,24);(1,2,4,3)(6,9)(10,12)(13,16,15,22,17,27,23,25)(14,19,18,24,21,26,20,28);(5,7,8,11)(6,10,9,12)(13,14)(15,18)(16,26)(17,21)(19,27)(20,23)(22,28)(24,25);(1,3,4,2)(13,15,17,23)(14,20,21,18)(16,22,27,25)(19,28,26,24)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724508);