Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#3199 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1119 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 1119 conjugacy classes of subgroups — too many to draw. Only its 138 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)(5,6,9,15,11,16,10,13)(7,17,18,32,21,33,20,36)(8,14,23,29,22,31,24,30)(12,25,26,35,28,34,27,19);(5,7)(6,12)(8,19)(9,20)(10,18)(11,21)(13,27)(14,17)(15,26)(16,28)(22,35)(23,34)(24,25)(29,32)(30,36)(31,33);(2,4)(6,13)(7,18)(8,22)(9,10)(14,29)(15,16)(19,25)(20,21)(26,27)(30,31)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72311946, #72311953, #72311969, … |
| maximal normal subgroup of | #72311946, #72311953, #72311969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55000);