Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #53977 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
545 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 545 conjugacy classes of subgroups — too many to draw. Only its 74 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:
(1,2,5,9,6,10,4,14)(3,11,12,7,13,8,15,16)(17,18)(19,29)(20,33,36,43)(21,27,22,26)(23,28)(24,31,41,37)(25,35)(30,42,34,39)(32,40)(38,44)(45,47)(46,48);(1,3)(2,7)(4,15)(5,12)(6,13)(8,9)(10,16)(11,14)(17,19)(18,24)(20,34)(21,31)(22,37)(23,32)(25,42)(26,29)(27,40)(28,41)(30,36)(33,48)(35,47)(38,46)(39,44)(43,45);(1,3)(2,8)(4,12)(5,15)(6,13)(7,9)(10,11)(14,16)(17,20,23,36)(18,25,28,44)(19,30,32,34)(21,35,22,38)(24,39,41,42)(26,33,27,43)(29,45,40,48)(31,46,37,47)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,e^c=ei,f^c=fgi,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, 40038);