Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2464 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,10,9)(7,8,12,11)(14,17)(15,19)(23,27)(25,28);(2,4)(5,7,10,12)(6,11,9,8)(13,14,16,17)(15,20,19,18);(1,2)(3,4)(13,14)(15,18)(16,17)(19,20)(21,22,24,26)(23,25,27,28);(5,8)(6,7)(9,12)(10,11)(13,14)(15,20)(16,17)(18,19)(21,23)(22,25)(24,27)(26,28);(5,9,10,6)(7,11,12,8)(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2762592);