Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C42 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C82 | C23 |
| socle | C22 | C2×#322 |
| prime radical | C23 | #322 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,4,9,5,10,16,25,6,8,17,22,18,23,31,32)(3,11,13,26,14,21,29,24,12,27,15,30,28,7,19,20)(33,34)(35,39)(36,41)(37,42)(38,40)(43,50)(44,51)(45,49)(46,54)(47,52)(48,53)(55,61)(56,59)(57,60)(58,62)(63,64);(1,3)(2,7,8,21)(4,15)(5,14)(6,12)(9,24,22,20)(10,11,23,27)(13,17)(16,19)(18,28)(25,30,32,26)(29,31)(33,35,36,43,37,44,46,55,38,45,47,56,48,57,58,63)(34,39,41,50,42,51,54,61,40,49,52,59,53,60,62,64);(2,8)(3,12)(9,22)(10,23)(13,15)(14,28)(19,29)(25,32)(34,40)(39,49)(41,52)(42,53)(50,59)(51,60)(54,62)(61,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bf,c^a=chi,e^a=eh,c^b=ch,d^b=dh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72231889, #72231894, #72231895, … |
| maximal normal subgroup of | #72231889, #72231894, #72231895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29414);