Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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 | |
3993 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)(9,10)(11,13)(12,14)(15,16)(18,19)(20,24);(6,8)(10,11)(12,16)(18,19)(22,23);(5,6)(7,8)(10,11)(14,15)(17,18,21,19)(20,22,24,23);(1,2,3,4)(9,11,13,10)(12,14,16,15)(17,19,21,18)(20,23,24,22);(9,12)(10,14)(11,15)(13,16)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^9|c^2=fg,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232212, #72232255, #72232259, … |
| maximal normal subgroup of | #72232212, #72232255, #72232259, … |
| maximal quotient of | #72330275, #72330341, #72330708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3042637);