Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500689 |
| 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 | #35533 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2282 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)(14,18)(19,27)(20,25)(21,28);(5,7,9,11)(6,10,12,8)(13,14,16,18)(15,21,22,28)(17,25,24,20)(19,26,27,23);(2,4)(5,8,9,10)(6,11,12,7)(13,15,16,22)(14,19,18,27)(17,26,24,23)(20,28,25,21);(2,4)(6,12)(7,11)(13,14)(15,19)(16,18)(17,20)(21,23)(22,27)(24,25)(26,28);(14,18)(19,27)(20,25)(21,28)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410962);