Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488553 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #61009 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2922 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
639 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 639 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 256 |
A minimal faithful permutation representation — 4 generators on 96 points:
(1,2,6,10)(3,7,13,20)(4,8,15,22)(5,16,17,9)(11,23,25,18)(12,26,27,19)(14,28,29,21)(24,30,32,31)(33,34)(35,47)(36,52)(37,56)(38,46)(39,45)(40,43)(41,49)(42,55)(44,59)(48,75)(50,81)(51,74)(53,71)(54,85)(57,72)(58,68)(60,70)(61,67)(62,69)(63,90)(64,80)(65,88)(66,84)(73,78)(76,93)(77,82)(79,95)(83,87)(86,92)(89,94)(91,96);(1,3)(2,7)(4,11)(5,12)(6,13)(8,18)(9,19)(10,20)(14,24)(15,25)(16,26)(17,27)(21,30)(22,23)(28,31)(29,32)(33,35)(34,41)(36,48)(37,57)(38,50)(39,62)(40,51)(42,63)(43,64)(44,69)(45,65)(46,72)(47,56)(49,59)(52,82)(53,76)(54,86)(55,78)(58,80)(60,88)(61,79)(66,93)(67,92)(68,81)(70,74)(71,91)(73,83)(75,85)(77,87)(84,90)(89,96)(94,95);(1,4)(2,8)(3,11)(5,14)(6,15)(7,18)(9,21)(10,22)(12,24)(13,25)(16,28)(17,29)(19,30)(20,23)(26,31)(27,32)(33,36,40,53)(34,42,43,67)(35,48,51,76)(37,54,60,84)(38,61,39,55)(41,63,64,92)(44,66,68,85)(45,71,46,52)(47,73,74,95)(49,77,80,96)(50,79,62,78)(56,83,70,94)(57,86,88,90)(58,89,59,87)(65,91,72,82)(69,93,81,75);(1,5,6,17)(2,9,10,16)(3,12,13,27)(4,14,15,29)(7,19,20,26)(8,21,22,28)(11,24,25,32)(18,30,23,31)(34,43)(36,53)(41,64)(44,68)(45,46)(47,74)(48,76)(54,84)(55,61)(56,70)(65,72)(69,81)(77,96)(78,79)(86,90)(87,89)Polycyclic presentation:
2^9|a^2=h,c^2=i,d^2=h,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395870);