Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2044 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14,19,16)(15,17,20,25)(18,21,28,26)(22,23,27,24);(5,7,8,12)(6,9,10,11)(13,15)(14,17)(16,25)(18,23)(19,20)(21,27)(22,26)(24,28);(2,4)(13,16,19,14)(15,22,20,27)(17,23,25,24)(18,21,28,26);(13,17)(14,20)(15,16)(18,27)(19,25)(21,24)(22,28)(23,26);(15,23)(17,27)(18,28)(20,24)(21,26)(22,25)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250261, #72256952, #72256973 |
| maximal normal subgroup of | #72250261, #72256952, #72256973 |
| maximal quotient of | #72264659, #72264756, #72264873, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481815);