Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #104437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #392 |
| commutator, Frattini, cosocle | C43 | C23 |
| LCS | G ▷ C43 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
401 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 512 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,10)(4,6)(5,15)(7,9)(8,11)(12,16)(13,14)(17,18,22,26)(19,27,20,24)(21,30,31,25)(23,29,32,28)(33,34,38,42)(35,43,46,47)(36,41,37,40)(39,45,48,44);(1,3,6,12)(2,8,4,14)(5,13,7,11)(9,16,15,10)(17,19,22,20)(18,23,26,32)(21,29,31,28)(24,30,27,25)(33,35,38,46)(34,39,42,48)(36,45,37,44)(40,47,41,43);(1,4,6,2)(3,10,12,16)(5,15,7,9)(8,11,14,13)(17,20,22,19)(18,24,26,27)(21,28,31,29)(23,25,32,30)(34,40)(36,37)(39,47)(41,42)(43,48)(44,45)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gi,d^2=g,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,e^b=ei,f^b=fgh,d^c=dh,e^c=ei,f^c=fgh53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 11704);