Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 |
| prime radical | C2×#3035 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2240 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)(5,6,8,9)(7,11,12,10)(13,14)(15,22)(16,27)(17,21)(18,20)(19,26)(23,25)(24,28);(6,9)(7,12)(13,15)(14,19)(16,25)(17,26)(18,24)(20,23)(21,22)(27,28);(2,4)(13,14)(15,23)(16,20)(17,21)(18,27)(19,24)(22,25)(26,28);(2,4)(5,7,8,12)(6,10,9,11)(13,16,17,18)(14,20,21,27)(15,24,26,25)(19,28,22,23);(13,16,17,18)(14,20,21,27)(15,25,26,24)(19,23,22,28)Polycyclic presentation:
2^9|a^2=fi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409406);