Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32894 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2846 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14694 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3658 conjugacy classes of subgroups; only the 416 normal subgroups are shown.
This group has 3658 conjugacy classes of subgroups — too many to draw. Only its 416 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:
(2,7,10,16)(8,18,21,25)(9,19,22,27)(11,23)(20,30,31,29)(24,32);(1,2,5,16,6,10,17,7)(3,8,12,25,13,21,26,18)(4,9,14,27,15,22,28,19)(11,20,24,30,23,31,32,29);(1,3)(2,8)(4,11)(5,12)(6,13)(7,18)(9,20)(10,21)(14,24)(15,23)(16,25)(17,26)(19,29)(22,31)(27,30)(28,32);(1,4)(2,9)(3,11)(5,14)(6,15)(7,19)(8,20)(10,22)(12,24)(13,23)(16,27)(17,28)(18,29)(21,31)(25,30)(26,32)Polycyclic presentation:
2^9|a^2=h,b^2=eh,e^2=i,b^a=be,c^a=cf,d^a=dg,e^b=ei,h^b=hi,g^c=gi,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72215925, #72219684, #72219691, … |
| maximal normal subgroup of | #72215925, #72219684, #72219691, … |
| maximal quotient of | #72343081 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299902);