Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #131330 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #3191 | C4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#433 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
525 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 525 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:
(1,2,5,10,6,9,20,24)(3,11,14,25,13,7,18,22)(4,15,17,26,16,8,19,23)(12,27,29,32,28,21,30,31)(33,34,38,44,39,45,60,74,40,46,61,75,59,73,37,43)(35,47,51,64,52,71,49,76,50,78,57,41,62,65,85,66)(36,42,54,68,55,69,53,67,56,70,87,94,86,93,58,72)(48,79,82,63,83,91,84,92,81,77,89,90,88,96,80,95);(1,3)(2,7)(4,12)(5,18)(6,13)(8,21)(9,11)(10,25)(14,20)(15,27)(16,28)(17,30)(19,29)(22,24)(23,32)(26,31)(33,35)(34,41)(36,48)(37,57)(38,49)(39,62)(40,50)(42,63)(43,71)(44,47)(45,76)(46,64)(51,60)(52,59)(53,82)(54,84)(55,88)(56,81)(58,89)(61,85)(65,74)(66,73)(67,91)(68,77)(69,95)(70,90)(72,96)(75,78)(79,94)(80,87)(83,86)(92,93);(1,4)(2,8)(3,12)(5,19)(6,16)(7,21)(9,15)(10,26)(11,27)(13,28)(14,30)(17,20)(18,29)(22,32)(23,24)(25,31)(33,36,40,56)(34,42,46,70)(35,48,50,81)(37,58,60,53)(38,54,61,87)(39,55,59,86)(41,63,64,90)(43,72,74,67)(44,68,75,94)(45,69,73,93)(47,77,78,79)(49,84,85,80)(51,82,57,89)(52,83,62,88)(65,91,71,96)(66,92,76,95)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=hi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dgi,f^c=fg56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38833);