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×#2404 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,7)(14,19)(16,23)(17,25)(22,27);(6,7)(10,12)(13,14,18,19)(15,16,20,23)(17,21,25,26)(22,24,27,28);(5,6,8,7)(9,10,11,12)(13,15)(14,16)(17,22)(18,20)(19,23)(21,24)(25,27)(26,28);(1,2)(3,4)(5,7,8,6)(13,16)(14,20)(15,19)(17,24)(18,23)(21,27)(22,26)(25,28);(13,17)(14,21)(15,22)(16,24)(18,25)(19,26)(20,27)(23,28)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=g,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, 825854);