Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219566 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2328 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23 | #540 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
444 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 96 points:
(1,2,7,11,5,10,6,22)(3,12,16,27,14,8,15,24)(4,17,21,26,19,31,20,9)(13,28,30,23,18,32,29,25)(33,34)(35,46)(36,51,56,79)(37,57)(38,61)(39,44)(40,45)(41,49)(42,55,69,54)(43,58)(47,63,73,77)(48,74)(50,65)(52,80,82,67)(53,83,86,68)(59,70)(60,87)(62,66)(64,76)(71,75)(72,91,92,95)(78,85,94,84)(81,89,96,93)(88,90);(1,3,6,15,5,14,7,16)(2,8,22,27,10,12,11,24)(4,18,20,30,19,13,21,29)(9,25,31,32,26,23,17,28)(33,35,38,49,40,50,39,62)(34,41,44,46,45,66,61,65)(36,52,55,47,56,82,54,73)(37,48,58,75,60,76,59,88)(42,67,79,63,69,80,51,77)(43,64,57,71,70,74,87,90)(53,81,85,72,86,96,84,92)(68,91,94,89,83,95,78,93);(1,4,7,21,5,19,6,20)(2,9,22,31,10,26,11,17)(3,13,16,30,14,18,15,29)(8,23,27,28,12,25,24,32)(33,36,40,56)(34,42)(35,47)(37,53,60,86)(38,54,39,55)(41,63,66,77)(43,68)(44,51)(45,69)(46,67,65,80)(48,72)(49,52)(50,73)(57,78)(58,84,59,85)(61,79)(62,82)(64,89,74,93)(70,83)(71,91,90,95)(75,81)(76,92)(87,94)(88,96);(2,10)(4,19)(8,12)(11,22)(13,18)(20,21)(24,27)(29,30)(33,37,39,59,40,60,38,58)(34,43,61,87,45,70,44,57)(35,48,62,88,50,76,49,75)(36,53,55,85,56,86,54,84)(41,64,65,90,66,74,46,71)(42,68,79,94,69,83,51,78)(47,72,82,96,73,92,52,81)(63,89,80,95,77,93,67,91)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=efh,d^2=ehi,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,d^c=di,g^c=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 126854);