Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #103845 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
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 | |
400 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(17,18,22,19)(20,23,28,25)(21,26,31,24)(27,30,32,29)(33,34,38,42)(35,43,46,47)(36,41,37,40)(39,45,48,44);(1,2,5,8)(3,7,10,16)(4,12,11,6)(9,14,13,15)(17,19,22,18)(20,27,28,32)(21,24,31,26)(23,29,25,30)(33,35,38,46)(34,39,42,48)(36,45,37,44)(40,47,41,43);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,20)(18,23)(19,25)(21,30)(22,28)(24,27)(26,32)(29,31)(34,40)(36,37)(39,47)(41,42)(43,48)(44,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,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=fgh,c^b=cf,d^b=dg,e^b=eg,f^b=fgh,d^c=dh,e^c=ei,f^c=fgh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 11164);