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 | |
2411 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,11)(13,14,19,23)(15,20,17,21)(16,26,22,18)(24,25,27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,17)(14,20,23,21)(16,27,22,24)(18,25,26,28);(2,4)(13,16,18,14,19,22,26,23)(15,24,25,21,17,27,28,20);(5,7,8,11)(6,10,9,12)(13,17,19,15)(14,21,23,20)(16,24,22,27)(18,28,26,25);(2,4)(6,9)(10,12)(15,17)(20,21)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=fi,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, 6590854);