Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149797 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×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 | C2×#493 | C22 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C2×C42 ◁ #2328 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
397 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 397 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:
(2,4)(5,6)(7,11)(8,19,21,12)(9,14,24,13)(10,15)(16,31,32,36)(17,27)(18,26,33,30)(20,35,34,25)(22,28,23,29);(1,2,3,4)(5,7)(6,11)(8,20)(9,18)(10,17)(12,25)(13,30)(14,26)(15,27)(16,22)(19,35)(21,34)(23,32)(24,33)(28,31)(29,36);(5,8)(6,12)(7,16)(9,23)(10,22)(11,25)(13,28)(14,19)(15,29)(17,20)(18,34)(21,24)(26,35)(27,36)(30,31)(32,33)Polycyclic presentation:
2^9|a^2=f,b^2=d,e^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=ei,e^c=ehi,f^c=fgh,g^c=gi,f^e=fhi,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72292975, #72292976, #72292977, … |
| maximal normal subgroup of | #72292975, #72292976, #72292977, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59451);