Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #917169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2397 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3968 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,7)(13,14,18,21)(15,16,23,19)(17,25,26,20)(22,27,28,24);(6,7)(10,12)(13,15,18,23)(14,19,21,16)(17,22,26,28)(20,27,25,24);(5,6,8,7)(9,10,11,12)(13,14)(15,19)(16,23)(17,20)(18,21)(22,27)(24,28)(25,26);(1,2)(3,4)(5,7,8,6)(13,16,18,19)(14,15,21,23)(17,24,26,27)(20,22,25,28);(13,17,18,26)(14,20,21,25)(15,22,23,28)(16,24,19,27)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 825861);