Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2403 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(6,8)(9,10,14,16)(11,13,12,15)(17,18,20,22)(19,23,24,21);(2,4)(9,11,14,12)(10,15,16,13)(17,18,20,22)(19,21,24,23);(1,2,3,4)(9,11,14,12)(10,15,16,13)(17,19)(18,21)(20,24)(22,23);(1,2,3,4)(5,6)(7,8)(9,12,14,11)(10,13,16,15)(17,18)(19,21)(20,22)(23,24);(1,2)(3,4)(9,13)(10,12)(11,16)(14,15)(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72300339, #72300385, #72300422, … |
| maximal normal subgroup of | #72300339, #72300385, #72300422, … |
| maximal quotient of | #72330500, #72330946, #72332935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3766342);