Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,20,19,16)(21,22)(23,25)(24,27)(26,28);(1,3,4,2)(6,8)(10,12)(14,18)(16,20)(21,22,25,23)(24,27,28,26);(7,11)(10,12)(14,18)(15,19)(22,23)(26,27);(5,7,9,11)(6,10,8,12)(13,15,17,19)(14,16,18,20)(21,23,25,22)(24,27,28,26);(5,8,9,6)(7,12,11,10)(13,16)(14,19)(15,18)(17,20)(21,24,25,28)(22,26,23,27)Polycyclic presentation:
2^9|a^2=i,b^2=gi,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 443559);