Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145318 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2034 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #469 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
235 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 235 conjugacy classes of subgroups — too many to draw. Only its 58 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)(3,12)(4,17)(5,22)(6,10)(7,11)(8,14)(9,20)(13,28)(15,24)(16,27)(18,23)(19,26)(21,31)(25,30)(29,32)(33,34,40,46)(35,41,50,66)(36,51,53,85)(37,57,61,73)(38,45,39,44)(42,67,69,54)(43,59,74,60)(47,76,77,96)(48,79,82,95)(49,75,62,65)(52,87,83,84)(55,86,56,72)(58,71,89,70)(63,91,92,78)(64,90,93,81)(68,88,94,80);(1,3,7,16)(2,8,11,24)(4,18,20,29)(5,15,6,14)(9,25,17,28)(10,27,22,12)(13,19,30,21)(23,31,32,26)(33,35)(34,41)(36,52)(37,48)(38,62)(39,49)(40,50)(42,68)(43,64)(44,75)(45,65)(46,66)(47,55)(51,84)(53,83)(54,88)(56,77)(57,79)(58,78)(59,90)(60,81)(61,82)(63,70)(67,80)(69,94)(71,92)(72,76)(73,95)(74,93)(85,87)(86,96)(89,91);(1,4,5,19,7,20,6,21)(2,9,10,26,11,17,22,31)(3,13,14,29,16,30,15,18)(8,23,12,25,24,32,27,28)(33,36,39,55,40,53,38,56)(34,42,45,70,46,69,44,71)(35,47,49,83,50,77,62,52)(37,58,60,67,61,89,59,54)(41,63,65,94,66,92,75,68)(43,72,57,85,74,86,73,51)(48,80,81,91,82,88,90,78)(64,87,79,96,93,84,95,76)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=dh,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dg,f^b=fgi,d^c=di,e^c=eghi,f^c=fi,h^c=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52535);