Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
872 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 872 conjugacy classes of subgroups — too many to draw. Only its 234 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 — 4 generators on 48 points:
(1,2,4,8,5,7,11,14)(3,6,9,13,10,12,15,16)(17,18)(19,27,30,23)(20,31,32,46)(21,26)(22,25)(24,33,37,47)(28,41)(29,40,44,36)(34,38)(35,43)(39,48)(42,45);(1,3,5,10)(2,6,7,12)(4,9,11,15)(8,13,14,16)(17,19,22,30)(18,23,25,27)(20,28,32,42)(21,29,34,44)(24,35,37,39)(26,36,38,40)(31,45,46,41)(33,43,47,48);(2,7)(4,11)(6,12)(9,15)(17,20)(18,24,25,37)(19,28)(21,33)(22,32)(23,35,27,39)(26,31,38,46)(29,43)(30,42)(34,47)(36,45,40,41)(44,48);(20,32)(24,37)(28,42)(31,46)(33,47)(35,39)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,d^c=dh,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97286);