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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2970 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)(5,6,10,9)(7,8,12,11)(13,14,17,16)(15,19,20,18)(22,26)(23,28);(5,7,10,12)(6,11,9,8)(13,15,17,20)(14,18,16,19)(21,22,25,26)(23,27,28,24);(6,9)(7,12)(13,14,17,16)(15,19,20,18)(21,23)(22,27)(24,26)(25,28);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,16,17,14)(15,18,20,19)(21,23)(22,27)(24,26)(25,28);(5,9,10,6)(7,11,12,8)(13,15)(14,18)(16,19)(17,20)(21,24,25,27)(22,28,26,23)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=fg,e^2=g,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, 3767110);