Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | #16249 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3257 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,9)(7,11,12,10)(14,17)(15,19)(21,22)(23,26)(24,25)(27,28);(2,4)(6,9)(7,12)(13,14,16,17)(15,20,19,18)(22,25)(23,28);(2,4)(7,12)(10,11)(14,17)(18,20)(21,23,24,28)(22,26,25,27);(2,4)(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,14)(15,20)(16,17)(18,19)(21,23)(22,27)(24,28)(25,26)Polycyclic presentation:
2^9|a^2=fgh,b^2=fhi,c^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3850249);