Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | #15258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3398 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)(7,9)(8,10)(11,12)(13,14)(15,20)(16,18)(17,19)(21,22,25,24)(23,27,28,26);(1,2)(3,4)(6,7)(10,11)(13,15)(14,16)(17,18)(19,20)(21,23)(22,26)(24,27)(25,28);(1,2,3,4)(8,12)(10,11)(15,18)(16,20)(21,22,25,24)(23,26,28,27);(5,7,9,6)(8,11,12,10)(13,14)(15,20)(16,18)(17,19)(21,24,25,22)(23,26,28,27);(1,2)(3,4)(5,8,9,12)(6,10,7,11)(13,16,17,20)(14,18,19,15)(21,23)(22,27)(24,26)(25,28)Polycyclic presentation:
2^9|a^2=fhi,c^2=fghi,d^2=fh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2456230);