Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10)(11,15)(12,14)(13,16)(18,22)(20,24);(9,10,12,14)(11,13,16,15)(17,18,21,22)(19,20,23,24);(5,6)(7,8)(11,16)(13,15)(18,22)(19,23);(1,2)(3,4)(5,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,23)(22,24);(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,24)(22,23)Polycyclic presentation:
2^9|b^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72231749, #72232078, #72232113, … |
| maximal normal subgroup of | #72231749, #72232078, #72232113, … |
| maximal quotient of | #72250271, #72278228, #72278491, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4218927);