Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2856 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(6,7)(8,12)(13,14)(15,20)(16,18)(17,19)(21,22,25,26)(23,27,24,28);(6,7)(10,11)(14,16)(17,20)(22,26)(23,24);(5,6,9,7)(8,10,12,11)(13,15)(14,17)(16,20)(18,19)(21,22)(23,27)(24,28)(25,26);(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,20,19,17)(21,23,25,24)(22,27,26,28);(5,8)(6,10)(7,11)(9,12)(13,17,18,20)(14,19,16,15)(21,24,25,23)(22,28,26,27)Polycyclic presentation:
2^9|a^2=i,c^2=fg,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 513949);