Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132521 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
578 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 578 conjugacy classes of subgroups — too many to draw. Only its 76 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:
(1,2,6,11,7,10,5,15)(3,12,13,9,4,8,14,16)(17,18,22,27,23,28,21,26)(19,29,33,24,32,39,31,38)(20,25,34,40,35,41,36,42)(30,44,45,48,46,43,47,37);(1,3,7,4)(2,8,10,12)(5,14,6,13)(9,15,16,11)(17,19)(18,24)(20,30)(21,33)(22,31)(23,32)(25,37)(26,39)(27,29)(28,38)(34,47)(35,46)(36,45)(40,43)(41,48)(42,44);(1,4,7,3)(2,9)(5,13,6,14)(8,15)(10,16)(11,12)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,37)(26,42)(27,40)(28,41)(29,43)(31,47)(32,46)(33,45)(38,48)(39,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,e^c=ei,f^c=fg,e^d=ei,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40172);