Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #108504 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #376 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1746 |
| prime radical | C23 | #433 |
| LCS | G ▷ C42 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #1450 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 96 points:
(1,2)(3,11,14,24)(4,8)(5,9)(6,10)(7,13,19,27)(12,23,26,31)(15,20)(16,21)(17,22)(18,25,29,32)(28,30)(33,34,37,42,38,44,54,66,39,43,55,65,56,67,81,91)(35,45,48,69,49,70,76,93,47,40,57,59,75,60,83,85)(36,41,51,62,53,63,80,88,52,64,79,89,71,90,94,96)(46,68,73,92,74,61,86,87,72,58,82,84,50,77,78,95);(1,3)(2,7,10,19)(4,12)(5,13)(6,14)(8,18,21,29)(9,11,22,24)(15,25)(16,26)(17,27)(20,23,30,31)(28,32)(33,35)(34,40,43,45)(36,50,52,74)(37,48)(38,49)(39,47)(41,61)(42,59,65,69)(44,60,67,70)(46,71,72,53)(51,78,79,86)(54,76)(55,57)(56,75)(58,63)(62,87)(64,77)(66,85,91,93)(68,90)(73,94,82,80)(81,83)(84,88)(89,95)(92,96);(1,4,5,15,6,16,17,28)(2,8,9,20,10,21,22,30)(3,12,13,25,14,26,27,32)(7,18,11,23,19,29,24,31)(33,36)(34,41)(35,46,47,72)(37,51)(38,53)(39,52)(40,58,45,68)(42,62)(43,64)(44,63)(48,73,57,82)(49,74,75,50)(54,80)(55,79)(56,71)(59,84,69,92)(60,77,70,61)(65,89)(66,88)(67,90)(76,86,83,78)(81,94)(85,95,93,87)(91,96)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=d,d^2=g,e^2=h,h^2=i,b^a=bd,f^a=fgi,c^b=chi,d^b=di,e^b=eg,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35432);