Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131660 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #447 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
500 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 500 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) = 96 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 96 points:
(1,2,6,12,7,10,5,11)(3,13,15,26,16,8,21,23)(4,17,19,9,20,25,18,24)(14,28,29,22,30,32,31,27)(33,34,39,45,38,47,63,43,40,46,64,73,62,44,37,61)(35,48,52,68,54,72,84,76,53,75,83,41,51,67,50,66)(36,55,58,69,60,71,89,87,59,74,88,90,57,42,56,70)(49,78,81,92,80,77,95,91,82,93,96,86,85,94,79,65);(1,3)(2,8)(4,14)(5,15)(6,21)(7,16)(9,22)(10,13)(11,23)(12,26)(17,32)(18,29)(19,31)(20,30)(24,27)(25,28)(33,35,40,53)(34,41,46,68)(36,49,59,82)(37,52,63,83)(38,51,62,54)(39,50,64,84)(42,65,71,91)(43,72,61,67)(44,66,47,76)(45,75,73,48)(55,86,74,92)(56,81,89,96)(57,80,60,85)(58,79,88,95)(69,93,90,78)(70,94,87,77);(1,4,7,20)(2,9,10,24)(3,14,16,30)(5,19,6,18)(8,22,13,27)(11,25,12,17)(15,31,21,29)(23,28,26,32)(33,36)(34,42)(35,49)(37,56)(38,60)(39,58)(40,59)(41,65)(43,69)(44,74)(45,70)(46,71)(47,55)(48,77)(50,79)(51,85)(52,81)(53,82)(54,80)(57,62)(61,90)(63,89)(64,88)(66,92)(67,78)(68,91)(72,93)(73,87)(75,94)(76,86)(83,96)(84,95)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=hi,e^2=g,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,f^b=fgh,h^b=hi,e^c=eg,f^c=fgi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38961);