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 |
| prime radical | C2×#3003 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3218 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(10,15)(11,18)(12,13)(16,23)(17,21)(19,20);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,21)(13,15)(14,17)(18,22)(19,23)(20,24);(2,6)(3,7)(9,11,13,18,14,19,12,20)(10,16,15,22,17,23,21,24);(3,7)(5,8);(11,19)(16,23)(18,20)(22,24)Polycyclic presentation:
2^9|c^2=gh,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213590, #72213818, #72213852, … |
| maximal normal subgroup of | #72213590, #72213818, #72213852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366654);