Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35223 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2313 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)(5,6,9,12)(7,8,11,10)(13,14)(15,18)(16,20)(17,19)(21,25)(22,27)(23,26)(24,28);(5,7,9,11)(6,10,12,8)(13,15,17,23)(14,18,19,26)(16,21,24,22)(20,25,28,27);(2,4)(5,8,9,10)(6,11,12,7)(15,21)(16,24)(18,25)(20,28)(22,23)(26,27);(2,4)(6,12)(7,11);(14,19)(18,26)(20,28)(25,27)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=f,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, 7410942);