Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #120908 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #382 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1832 |
| prime radical | C23 | #433 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
152 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)(3,11,14,7)(4,15,17,30)(5,10)(6,9)(8,16,21,31)(12,25)(13,24,28,20)(18,22)(19,26)(23,32)(27,29)(33,34,38,44,40,43,61,71,39,45,60,72,37,57,59,85)(35,46,49,74,48,41,62,64,50,66,77,90,58,65,86,89)(36,51,55,79,56,78,84,95,54,42,67,70,53,69,83,92)(47,68,76,91,52,80,82,96,75,73,93,94,81,63,87,88);(1,3,5,13,6,14,18,28)(2,7,10,20,9,11,22,24)(4,12,16,26,17,27,31,32)(8,19,15,29,21,23,30,25)(33,35)(34,41)(36,52,54,81)(37,58)(38,49)(39,50)(40,48)(42,68,51,73)(43,66)(44,64)(45,65)(46,57)(47,56,75,53)(55,82,67,87)(59,86)(60,77)(61,62)(63,69,80,78)(70,91,79,94)(71,90)(72,89)(74,85)(76,84,93,83)(88,92,96,95);(1,4)(2,8,9,21)(3,12)(5,16)(6,17)(7,19,11,23)(10,15,22,30)(13,26)(14,27)(18,31)(20,29,24,25)(28,32)(33,36)(34,42,45,51)(35,47,50,75)(37,53)(38,55)(39,54)(40,56)(41,63)(43,69,57,78)(44,70,72,79)(46,73)(48,52,58,81)(49,76,77,93)(59,83)(60,67)(61,84)(62,82,86,87)(64,88)(65,80)(66,68)(71,92,85,95)(74,94)(89,96)(90,91)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=dh,d^2=i,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ch,d^b=di,f^b=fgi,e^c=ei,f^c=fgi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42838);