Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1464 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
534 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 534 conjugacy classes of subgroups — too many to draw. Only its 114 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:
(1,2,4,3)(5,6)(7,11,16,23)(8,13)(9,14)(10,12)(15,28,18,22)(17,32,31,24)(19,25)(20,27)(21,26)(29,36)(30,34,33,35);(5,7,8,17,9,18,20,33,10,16,19,31,21,15,29,30)(6,11,13,24,14,22,27,34,12,23,25,32,26,28,36,35);(1,3,4,2)(6,12)(7,15,16,18)(8,19)(11,22,23,28)(14,26)(17,33,31,30)(20,29)(24,35,32,34)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,f^2=h,h^2=i,b^a=bd,c^a=ci,d^a=di,f^a=fg,c^b=ch,d^b=dg,e^b=ei,g^b=gi,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#128379, #72217918 |
| maximal normal subgroup of | C2×#128379, #72217918 |
| maximal quotient of | #72216852, C2×#128379 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35575);