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 | #23426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,9)(7,11,12,10)(13,14,17,21)(15,22,24,19)(16,27,18,20)(23,25,28,26);(6,9)(7,12)(13,15)(14,19)(16,26)(17,24)(18,25)(20,23)(21,22)(27,28);(2,4)(13,14)(15,23)(16,20)(17,21)(18,27)(19,25)(22,26)(24,28);(2,4)(5,7,8,12)(6,10,9,11)(15,24)(19,22)(23,28)(25,26);(13,16,17,18)(14,20,21,27)(15,25,24,26)(19,28,22,23)Polycyclic presentation:
2^9|a^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,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, 7409415);