Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442584 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,15)(14,19)(18,21)(20,23)(22,24);(10,15)(11,20)(12,22)(13,21)(16,17)(18,19);(2,5)(6,8)(9,11,21,22,14,12,13,20)(10,16,18,24,19,17,15,23);(9,12,21,20,14,11,13,22)(10,17,18,23,19,16,15,24);(1,3,4,7)(2,6,5,8)(11,12)(16,17)(20,22)(23,24)Polycyclic presentation:
2^9|c^2=hi,d^2=hi,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237280, #72238931, #72238934, … |
| maximal normal subgroup of | #72237280, #72238931, #72238934, … |
| maximal quotient of | #72238763, C2×#6467260, #72239014, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374456);