Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
4099 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)(11,12)(13,15)(14,16)(18,21)(19,23);(5,6,7,8)(9,11,14,15)(10,13,16,12)(19,23)(22,24);(5,6)(7,8)(9,12,14,13)(10,11,16,15)(17,18)(19,22)(20,21)(23,24);(1,2,3,4)(9,13,14,12)(10,15,16,11)(17,19,20,23)(18,22,21,24);(9,10)(11,13)(12,15)(14,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|b^2=h,c^2=f,d^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72256756, #72256768, #72256791, … |
| maximal normal subgroup of | #72256756, #72256768, #72256791, … |
| maximal quotient of | #72298146, #72298358 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3372053);