Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14811 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,11)(7,8,12,10)(13,14,18,20)(15,16,19,17)(22,25)(23,27);(5,7)(6,10)(8,11)(9,12)(13,15,18,19)(14,16,20,17)(22,25)(26,28);(5,6)(7,8)(9,11)(10,12)(13,16,18,17)(14,15,20,19)(21,22,24,25)(23,26,27,28);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,14)(15,16)(17,19)(18,20);(5,8,9,10)(6,7,11,12)(13,17,18,16)(14,19,20,15)(21,23,24,27)(22,26,25,28)Polycyclic presentation:
2^9|a^2=fgh,b^2=gh,c^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1167226);