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 | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,10,12)(7,9,8,11)(13,14,18,16)(15,20,19,17)(21,22,25,26)(23,24,27,28);(5,7,10,8)(6,11,12,9)(13,15)(14,17)(16,20)(18,19)(21,22)(23,24)(25,26)(27,28);(6,12)(9,11)(15,19)(17,20)(22,26)(24,28);(1,2)(3,4)(5,8,10,7)(6,9,12,11)(13,16,18,14)(15,17,19,20)(21,23)(22,24)(25,27)(26,28);(5,9)(6,8)(7,12)(10,11)(13,17)(14,19)(15,16)(18,20)(21,24)(22,27)(23,26)(25,28)Polycyclic presentation:
2^9|a^2=f,b^2=fh,d^2=gh,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, 3774083);