Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488549 |
| 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 | C2×#41 | C2×D4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2922 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
627 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 627 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)(33,34,40,43)(35,47,52,77)(36,53,54,75)(37,57,62,74)(38,46,39,45)(41,64,66,49)(42,63,70,56)(44,60,71,61)(48,79,80,96)(50,72,51,78)(55,73,59,69)(58,67,84,68)(65,88,95,91)(76,85,93,82)(81,86,83,94)(87,92,90,89);(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)(33,35,40,52)(34,41,43,66)(36,48,54,80)(37,58,62,84)(38,51,39,50)(42,65,70,95)(44,72,71,78)(45,68,46,67)(47,74,77,57)(49,60,64,61)(53,86,75,94)(55,88,59,91)(56,85,63,82)(69,96,73,79)(76,87,93,90)(81,92,83,89);(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)(34,42)(35,48)(37,59)(38,63)(39,56)(40,54)(41,65)(43,70)(44,73)(45,75)(46,53)(47,76)(49,83)(50,85)(51,82)(52,80)(55,62)(57,90)(58,91)(60,89)(61,92)(64,81)(66,95)(67,86)(68,94)(69,71)(72,79)(74,87)(77,93)(78,96)(84,88);(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)(36,54)(41,66)(44,71)(45,46)(47,77)(48,80)(55,59)(56,63)(57,74)(67,68)(72,78)(81,83)(82,85)(88,91)(89,92)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=hi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,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=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 395867);