Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2668 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3206 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(6,9)(10,12)(13,14)(15,19)(16,20)(17,18);(1,2,4,3)(14,18)(15,20)(22,25)(26,28);(2,3)(14,18)(16,19)(21,22)(23,26)(24,25)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,19)(18,20)(21,23,24,27)(22,26,25,28)Polycyclic presentation:
2^9|b^2=fg,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 978767);