Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
4417 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)(11,13)(12,16)(14,15)(17,18)(19,22)(20,24)(21,23);(1,2)(3,4)(10,15)(11,12)(17,19,21,24)(18,22,23,20);(1,2,3,4)(9,11,14,12)(10,13,15,16)(17,19)(18,20)(21,24)(22,23);(5,6,7,8)(9,12,14,11)(10,13,15,16)(17,19)(18,20)(21,24)(22,23);(1,2)(3,4)(9,13)(10,11)(12,15)(14,16)(17,20)(18,19)(21,22)(23,24)Polycyclic presentation:
2^9|b^2=fh,c^2=f,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72278043, #72278319, #72293937, … |
| maximal normal subgroup of | #72278043, #72278319, #72293937, … |
| maximal quotient of | #72278117, #72278411, #72278994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2216807);