Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #348158 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3460 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
841 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 841 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,5,8)(3,6,10,13)(4,7,11,14)(9,15,16,12)(17,18,23,28)(19,29,33,40)(20,34,37,44)(21,27,22,26)(24,32,41,31)(25,38,43,36)(30,42,35,39)(45,47,48,46);(17,19,22,32,23,33,21,31)(18,24,27,40,28,41,26,29)(20,35,38,46,37,30,36,47)(25,42,44,45,43,39,34,48);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,20)(18,25)(19,30)(21,36)(22,38)(23,37)(24,39)(26,34)(27,44)(28,43)(29,45)(31,46)(32,47)(33,35)(40,48)(41,42);(1,4,5,11)(2,7,8,14)(3,9,10,16)(6,12,13,15)Polycyclic presentation:
2^9|a^2=hi,b^2=eh,d^2=i,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,c^b=ch,f^c=fh,g^c=gi,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 255365);