Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #120209 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1441 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
597 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 597 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,6)(4,11,12,7)(5,8)(9,13,16,15)(10,14)(17,18)(19,28)(20,33,36,47)(21,37)(22,26)(23,27)(24,30)(25,35,40,34)(29,43,46,38)(31,39)(32,42)(41,45,48,44);(1,3)(2,6)(4,9)(5,10)(7,15)(8,14)(11,13)(12,16)(17,19,21,30,23,32,22,31)(18,24,26,28,27,39,37,42)(20,29,34,44,36,46,35,45)(25,38,33,41,40,43,47,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,20,21,34,23,36,22,35)(18,25,26,33,27,40,37,47)(19,29,30,44,32,46,31,45)(24,38,28,41,39,43,42,48)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,e^b=ei,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 27405);