Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149931 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2127 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | #25734 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
664 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 664 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,11,17,25)(8,20,21,12)(9,15,23,13)(10,14)(16,24,30,35)(18,26,19,28)(22,29)(27,34)(31,33,32,36);(5,7)(6,11,15,25)(8,16)(9,17,23,19)(10,18)(12,24,27,36)(13,28,14,26)(20,33,29,35)(21,31)(22,32,34,30);(2,4)(5,8)(6,12)(7,16)(9,22)(10,21)(11,24)(13,29)(14,20)(15,27)(17,32)(18,31)(19,30)(23,34)(25,36)(26,33)(28,35)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dgh,e^b=ei,f^b=fgh,g^b=ghi,h^b=hi,d^c=di,f^c=fi,f^d=fhi,g^f=gi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72285058, #72285394 |
| maximal normal subgroup of | #72285058, #72285394 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59946);