Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12707 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1668 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1668 conjugacy classes of subgroups — too many to draw. Only its 202 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:
(5,6,10,15,11,16,8,13)(7,17,18,34,20,33,25,32)(9,22,21,31,24,14,23,30)(12,26,29,36,28,35,27,19);(1,2,3,4)(5,7)(6,12,16,28)(8,18)(9,19)(10,25)(11,20)(13,27,15,29)(14,17,22,33)(21,35)(23,26)(24,36)(30,34,31,32);(5,8,11,10)(6,13,16,15)(7,18,20,25)(9,23,24,21)(12,27,28,29)(14,31,22,30)(17,32,33,34)(19,26,36,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,e^b=ei,f^b=fhi,d^c=di,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72210061, #72210062, #72210063, … |
| maximal normal subgroup of | #72210061, #72210062, #72210063, … |
| maximal quotient of | C2×#146482, #72267888 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53678);