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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2418 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)(7,12)(8,10)(9,11)(14,17)(18,19)(21,22,25,28)(23,24,27,26);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,14,16,17)(15,19,20,18)(23,27)(24,26);(1,2)(3,4)(6,11)(10,12)(14,17)(18,19)(21,23,25,27)(22,26,28,24);(5,8,9,7)(6,12,11,10)(13,15,16,20)(14,18,17,19)(21,24,25,26)(22,27,28,23);(5,6)(7,12)(8,10)(9,11)(13,14)(15,19)(16,17)(18,20)(21,22)(23,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gi,d^2=fgi,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, 3920233);