Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #616960 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15124 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3136 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)(7,9)(8,11)(10,12)(14,17)(15,19)(23,27)(26,28);(6,7)(10,11)(13,14,16,17)(15,18,19,20)(22,25)(26,28);(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,27)(25,28);(5,8,9,12)(6,10,7,11)(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^9|b^2=g,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 533472);