Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391639 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14649 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
745 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 745 conjugacy classes of subgroups — too many to draw. Only its 188 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:
(1,2,6,12,7,11,20,27)(3,8,15,22,14,21,30,31)(4,9,17,24,16,23,19,28)(5,10,18,26,13,25,29,32);(1,3)(2,8)(4,13,16,5)(6,15)(7,14)(9,10,23,25)(11,21)(12,22)(17,29,19,18)(20,30)(24,26,28,32)(27,31);(1,4)(2,9,11,23)(3,13,14,5)(6,19)(7,16)(8,10)(12,28,27,24)(15,18,30,29)(17,20)(21,25)(22,32)(26,31);(1,5,7,13)(2,10,11,25)(3,4)(6,18,20,29)(8,9)(12,26,27,32)(14,16)(15,17)(19,30)(21,23)(22,24)(28,31)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=ef,d^2=gi,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
| maximal subgroup of | #72284785, #72284899, #72284914, … |
| maximal normal subgroup of | #72284785, #72284899, #72284914, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298837);