Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2419 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,12)(9,11)(13,14)(15,19)(16,21)(17,20)(18,22)(23,25)(24,28)(26,27);(2,3)(5,7,10,6)(8,12,9,11)(13,15)(14,19)(16,17)(18,24)(20,21)(22,28)(23,27)(25,26);(1,2,4,3)(14,20)(15,24)(17,25)(19,27)(21,28)(22,23);(5,8,10,9)(6,11,7,12)(13,16)(14,21)(15,25)(17,24)(18,26)(19,23)(20,28)(22,27);(1,3,4,2)(5,9,10,8)(6,11,7,12)(14,22)(15,24)(16,26)(20,23)Polycyclic presentation:
2^9|b^2=f,c^2=hi,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718864);