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 | #32902 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,25)(16,21)(17,22)(18,23)(19,24)(20,27)(26,28);(6,9)(10,12)(13,15,18,27)(14,20,23,25)(16,26,19,17)(21,22,24,28);(2,4)(13,16)(14,21)(17,26)(18,19)(22,28)(23,24);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,16)(18,26)(19,27)(20,24)(21,25)(23,28);(15,26)(16,19)(17,27)(20,22)(21,24)(25,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72256952, #72256956, #72256992 |
| maximal normal subgroup of | #72256952, #72256956, #72256992 |
| maximal quotient of | #72264865, #72265160, #72265189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483360);