Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133904 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #2309 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
974 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 974 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,21,25,31)(12,26,28,34)(13,19,14,18)(15,20)(16,23)(17,27)(22,36)(24,32)(29,33)(30,38)(35,39,40,37);(1,3,4,8)(2,5,6,7)(9,11)(10,16,20,32)(12,22)(13,23)(14,24)(15,25)(17,30,33,36)(18,21,19,31)(26,35,34,40)(27,37)(28,38)(29,39);(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,26)(19,34)(20,33)(21,35)(23,37)(24,39)(25,38)(31,40)(32,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,f^b=fgi,e^c=ei,f^c=fhi,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72285179, #72285208, #72285255, … |
| maximal normal subgroup of | #72285179, #72285208, #72285255, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41100);