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)(33,34,38,45,40,46,60,43,39,44,62,73,59,72,37,58)(35,47,50,65,52,71,83,76,51,74,85,41,63,66,49,67)(36,53,55,75,57,88,54,68,56,42,61,69,89,70,90,87)(48,78,80,86,82,95,79,96,81,77,84,64,92,93,91,94);(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)(33,35,39,51)(34,41,44,65)(36,48,56,81)(37,50,60,85)(38,49,62,83)(40,63,59,52)(42,64,53,86)(43,71,58,66)(45,74,73,47)(46,76,72,67)(54,84,90,80)(55,91,61,79)(57,92,89,82)(68,93,87,95)(69,77,75,78)(70,96,88,94);(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)(34,42)(35,48)(37,54)(38,61)(39,56)(40,57)(41,64)(43,68)(44,53)(45,75)(46,70)(47,77)(49,79)(50,84)(51,81)(52,82)(55,62)(58,87)(59,89)(60,90)(63,92)(65,86)(66,95)(67,94)(69,73)(71,93)(72,88)(74,78)(76,96)(80,85)(83,91)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=hi,f^2=ghi,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, 38835);