Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #249355 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35347 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2169 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 2169 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,7)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,24)(21,23);(1,2,5,4)(3,6,8,7)(14,23)(17,24);(1,3)(2,6)(4,7)(5,8)(9,11,13,18)(10,14,16,23)(12,21,22,19)(15,24,20,17);(1,4,5,2)(3,7,8,6)(11,18)(12,22)(14,23)(15,20)Polycyclic presentation:
2^9|b^2=hi,c^2=i,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^d=fh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72227660, #72227670, #72227681, … |
| maximal normal subgroup of | #72227660, #72227670, #72227681, … |
| maximal quotient of | #72232570, #72234720, #72234723, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 156585);