Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2331 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)(7,8)(9,11)(10,12)(15,22)(19,26)(23,28)(25,27);(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,18)(17,20)(21,24)(22,26)(23,25)(27,28);(2,4)(5,8,9,10)(6,7,11,12)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(2,4)(6,11)(7,12)(13,15)(14,19)(16,23)(17,22)(18,25)(20,26)(21,27)(24,28);(14,20)(15,22)(18,21)(23,28)Polycyclic presentation:
2^9|c^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408413);