Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129412 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #35676 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1720 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 146 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:
(5,6)(7,15,18,29)(8,19,21,11)(9,13)(10,14)(12,17,27,23)(16,32)(20,28,35,25)(22,30)(24,33)(26,36)(31,34);(1,2)(3,4)(6,11)(9,22)(12,26)(13,28)(14,19)(15,31)(17,23)(20,35)(24,27)(25,30)(29,32)(33,36);(2,4)(5,7,10,18)(6,12,14,27)(8,16,21,34)(9,23,22,17)(11,24,19,26)(13,29,30,15)(20,36,35,33)(25,32,28,31)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,g^b=gh,e^c=ei,g^c=gi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72267986, #72267995 |
| maximal normal subgroup of | #72267986, #72267995 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54774);