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 | #16291 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2599 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,12)(8,11)(9,10)(14,19)(17,20)(21,22,24,25)(23,27,28,26);(5,7,10,8)(6,11,9,12)(13,14)(15,17)(16,20)(18,19)(22,25)(26,27);(5,7,10,8)(6,12,9,11)(13,15,18,16)(14,20,19,17)(21,23,24,28)(22,26,25,27);(1,2)(3,4)(5,8,10,7)(6,12,9,11)(13,16,18,15)(14,20,19,17)(21,22)(23,26)(24,25)(27,28);(5,9,10,6)(7,12,8,11)(13,17,18,20)(14,15,19,16)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgh,c^2=f,d^2=fg,e^2=fg,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, 914360);