Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2166 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)(13,14,18,20)(15,22,24,19)(16,26,21,17)(23,28,27,25);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,27)(17,25)(18,24)(20,22)(21,23)(26,28);(1,2,4,3)(6,9)(10,12)(13,16,17,14,18,21,26,20)(15,23,25,22,24,27,28,19);(5,7,8,11)(6,10,9,12)(14,20)(15,24)(16,21)(25,28);(1,3,4,2)(13,17,18,26)(14,16,20,21)(15,25,24,28)(19,27,22,23)Polycyclic presentation:
2^9|a^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,d^b=di,e^b=efh,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, 6724420);