Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143205 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #447 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
407 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(6,10)(7,11)(8,12)(13,14,20,26)(15,27,32,43)(16,23,34,40)(17,22,33,41)(18,25,19,24)(21,35,39,29)(28,37,44,38)(30,36,31,42);(1,2)(3,4)(5,6,9,7)(8,11,12,10)(13,15,20,32)(14,21,26,39)(16,29,34,35)(17,28,33,44)(18,31,19,30)(22,27,41,43)(23,36,40,42)(24,38,25,37);(2,4)(5,7,9,6)(8,10,12,11)(13,16,19,33,20,34,18,17)(14,22,25,40,26,41,24,23)(15,28,31,35,32,44,30,29)(21,36,38,43,39,42,37,27)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=g,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,e^c=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50496);