Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #134747 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1812 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #8609 |
| prime radical | #532 | D4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #2113 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
295 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,8,14,7,29)(3,15,19,43,21,44,20,52)(4,22,24,39,25,55,16,11)(5,12,27,37,26,10,28,36)(9,30,33,48,34,45,41,17)(18,42,50,63,49,40,51,62)(23,54,46,38,56,53,57,35)(31,58,59,47,32,61,60,64);(1,3,16,45,7,20,25,48,8,21,24,30,6,19,4,17)(2,9,22,43,13,33,39,44,14,34,55,52,29,41,11,15)(5,18,46,61,27,50,56,64,26,49,57,58,28,51,23,47)(10,31,53,63,37,60,38,42,12,32,54,62,36,59,35,40);(2,10)(5,26)(9,31)(11,35)(12,14)(13,37)(15,40)(18,49)(22,53)(23,56)(27,28)(29,36)(32,34)(33,60)(38,39)(41,59)(42,44)(43,63)(46,57)(47,64)(50,51)(52,62)(54,55)(58,61)Polycyclic presentation:
2^9|a^2=f,b^2=dg,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,e^c=ei,f^c=fhi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72287306, #72287315, #72287362, … |
| maximal normal subgroup of | #72287306, #72287315, #72287362, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41973);