Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476491 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17091 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1347 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 1347 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,22,7,10,6,11)(3,12,15,25,16,8,23,26)(4,17,19,9,21,29,20,28)(13,27,31,30,14,32,18,24);(1,3)(2,8)(4,18)(5,23)(6,15)(7,16)(9,27)(10,12)(11,26)(13,20)(14,19)(17,30)(21,31)(22,25)(24,29)(28,32);(1,4)(2,9,10,28)(3,13)(5,19)(6,20)(7,21)(8,24,12,30)(11,17,22,29)(14,16)(15,31)(18,23)(25,32,26,27);(1,4)(2,9)(3,14)(5,20)(6,19)(7,21)(8,24)(10,28)(11,29)(12,30)(13,16)(15,31)(17,22)(18,23)(25,27)(26,32)Polycyclic presentation:
2^9|a^2=ef,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,g^c=gi,f^d=fi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72287334, #72287341, #72287354, … |
| maximal normal subgroup of | #72287334, #72287341, #72287354, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383687);