Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391095 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14609 |
| 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 | |
884 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 884 conjugacy classes of subgroups — too many to draw. Only its 196 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)(3,11)(4,8,17,24)(5,18,19,9)(6,10)(7,14)(12,22,15,20)(13,25,27,21)(16,23)(26,31)(28,32)(29,30);(1,3)(2,7,10,11)(4,15,17,12)(5,13,19,27)(6,14)(8,22)(9,21)(16,28)(18,25)(20,24)(23,32,30,31)(26,29);(1,4)(2,8,10,24)(3,12)(5,16)(6,17)(7,20,11,22)(9,23,18,30)(13,26)(14,15)(19,29)(21,31,25,32)(27,28);(1,5)(2,9)(3,13,14,27)(4,16)(6,19)(7,21,11,25)(8,23)(10,18)(12,26,15,28)(17,29)(20,31,22,32)(24,30)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fh,d^2=g,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237848, #72237849, #72237852, … |
| maximal normal subgroup of | #72237848, #72237849, #72237852, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298291);