Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #219590 |
| 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)(38,62)(39,44)(40,45)(41,49)(42,55,70,54)(43,59)(47,64,75,77)(48,71)(50,66)(52,80,82,68)(53,83,87,69)(58,65)(60,72)(61,89)(63,67)(73,76)(74,93,90,94)(78,86,96,85)(81,92,84,95)(88,91);(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)(33,35,38,49,40,50,39,63)(34,41,44,46,45,67,62,66)(36,52,55,47,56,82,54,75)(37,58,59,91,61,48,60,76)(42,68,79,64,70,80,51,77)(43,71,57,88,72,65,89,73)(53,84,86,90,87,81,85,74)(69,94,96,95,83,93,78,92);(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)(37,53,61,87)(38,54,39,55)(41,64,67,77)(43,69)(44,51)(45,70)(46,68,66,80)(48,74)(49,52)(50,75)(57,78)(58,90)(59,85,60,86)(62,79)(63,82)(65,92,71,95)(72,83)(73,93,88,94)(76,81)(84,91)(89,96);(2,10)(4,19)(8,12)(11,22)(13,18)(20,21)(24,27)(29,30)(33,37,38,59,40,61,39,60)(34,43,44,57,45,72,62,89)(35,48,49,76,50,58,63,91)(36,53,54,85,56,87,55,86)(41,65,46,73,67,71,66,88)(42,69,51,78,70,83,79,96)(47,74,52,81,75,90,82,84)(64,92,68,93,77,95,80,94)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=efh,d^2=ei,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,d^b=dh,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, 126865);