Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #103989 |
| 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 | |
414 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 88 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)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32)(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)(17,18,21,24)(19,23,26,32)(20,28,27,22)(25,30,29,31)(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)(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31)(34,40)(36,37)(39,47)(41,42)(43,48)(44,45)Polycyclic presentation:
2^9|a^2=i,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,c^b=cf,d^b=dg,e^b=eh,f^b=fgh,d^c=dh,e^c=ei,f^c=fgh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 11224);