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 | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
848 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 848 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)(3,7)(4,8,15,21)(5,9)(6,10)(11,24,26,17)(12,18)(13,19)(14,20,28,31)(16,23)(22,27)(25,30,32,29);(1,3)(2,7)(4,11,15,26)(5,12,16,27)(6,13)(8,17,21,24)(9,22,23,18)(10,19)(14,25)(20,30)(28,32)(29,31);(1,4)(2,8,10,21)(3,11,13,26)(5,14,16,28)(6,15)(7,17)(9,20)(12,25)(18,29,22,30)(19,24)(23,31)(27,32);(1,5)(2,9)(3,12,13,27)(4,14,15,28)(6,16)(7,18,19,22)(8,20,21,31)(10,23)(11,25)(17,29)(24,30)(26,32)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=ef,d^2=g,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 | #72237839, #72237840, #72237842, … |
| maximal normal subgroup of | #72237839, #72237840, #72237842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298835);