Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #329 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | C2×#376 | C22 |
| socle | C22 | #1270 |
| prime radical | C24 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C2×#376 ▷ C22×C4 ▷ C2 ▷ C1 | |
400 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 2 generators on 36 points:
(1,2,3,4)(5,6)(7,16)(8,12)(9,13)(10,15)(11,17)(14,23)(18,31)(19,34)(20,24)(21,29)(22,30)(25,32)(26,35)(27,33)(28,36);(5,7)(6,11,13,24,15,25,14,26)(8,20,21,35)(9,19)(10,18)(12,27,28,31,29,34,30,16)(17,22,32,36)(23,33)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,e^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efg,g^a=gi,h^a=hi,c^b=cg,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#93376, #72213056, #72213057, … |
| maximal normal subgroup of | C2×#93376, #72213056, #72213057, … |
| maximal quotient of | #72210220, #72210668, C2×#93376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 572);