Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142544 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator, Frattini, cosocle | C2×#117 | C23 |
| prime radical | #23988 | C2 |
| LCS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
575 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 575 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,22,27)(19,28,31,44)(20,25,33,41)(21,24,34,26)(23,35,37,29)(30,42,46,43)(32,39,47,40)(36,45,38,48);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14)(17,19)(18,23)(20,29)(21,30)(22,31)(24,38)(25,28)(26,36)(27,37)(32,45)(33,35)(34,46)(39,42)(40,43)(41,44)(47,48);(6,15)(9,16)(18,24)(21,34)(23,36)(25,39)(26,27)(28,42)(29,35)(30,46)(32,47)(37,38)(40,41)(43,44)Polycyclic presentation:
2^9|a^2=gi,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,h^b=hi,d^c=dh,e^c=ei,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72284041, #72284113, #72284138, … |
| maximal normal subgroup of | #72284041, #72284113, #72284138, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49774);