Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365522 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #61897 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,14,27)(4,9,19,26)(5,11,6,10)(8,20,22,16)(15,29,30,24)(17,32,31,23)(18,28,21,25)(33,34,39,44)(35,40,47,55)(36,49,51,58)(37,43,38,42)(41,52,57,50)(45,59,61,64)(46,56,48,54)(53,62,63,60);(1,3,7,14)(2,8,12,22)(4,17,19,31)(5,16,6,20)(9,24,26,29)(10,13,11,27)(15,21,30,18)(23,28,32,25)(33,35)(34,40)(36,45)(37,46)(38,48)(39,47)(41,53)(42,54)(43,56)(44,55)(49,59)(50,60)(51,61)(52,62)(57,63)(58,64);(3,14)(8,22)(13,27)(15,30)(16,20)(17,31)(23,32)(24,29)(33,36,39,51)(34,41,44,57)(35,45,47,61)(37,50,38,52)(40,53,55,63)(42,49,43,58)(46,60,48,62)(54,59,56,64);(1,4)(2,9)(3,15)(5,18)(6,21)(7,19)(8,23)(10,25)(11,28)(12,26)(13,29)(14,30)(16,31)(17,20)(22,32)(24,27)(33,37,39,38)(34,42,44,43)(35,46,47,48)(36,50,51,52)(40,54,55,56)(41,49,57,58)(45,60,61,62)(53,59,63,64);(1,5,7,6)(2,10,12,11)(3,16,14,20)(4,18,19,21)(8,13,22,27)(9,25,26,28)(15,31,30,17)(23,29,32,24)(35,47)(40,55)(45,61)(46,48)(53,63)(54,56)(59,64)(60,62)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ei,f^b=fh,g^c=gi,f^d=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272849);