Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6350717 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62968 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,12)(4,9)(5,10)(6,19)(7,11)(8,14)(13,26)(15,22)(16,24)(17,25)(18,29)(20,23)(21,28)(27,32)(30,31)(33,34,39,45)(35,42,49,56)(36,50,54,59)(37,43,38,44)(40,47,57,48)(41,53,60,52)(46,62,51,55)(58,64,61,63);(1,3,7,15)(2,8,11,23)(4,13,16,27)(5,14,6,20)(9,21,24,31)(10,22,19,12)(17,28,18,30)(25,32,29,26)(33,35)(34,40)(36,46)(37,47)(38,48)(39,49)(41,55)(42,43)(44,56)(45,57)(50,58)(51,54)(52,63)(53,64)(59,61)(60,62);(1,4,7,16)(2,9,11,24)(3,13,15,27)(5,17,6,18)(8,21,23,31)(10,25,19,29)(12,26,22,32)(14,28,20,30)(33,36)(34,41)(35,46)(37,52)(38,53)(39,54)(40,55)(42,61)(43,59)(44,50)(45,60)(47,63)(48,64)(49,51)(56,58)(57,62);(4,16)(9,24)(13,27)(17,18)(21,31)(25,29)(26,32)(28,30)(33,35)(34,42)(36,51)(37,48)(38,47)(39,49)(40,43)(41,58)(44,57)(45,56)(46,54)(50,55)(52,63)(53,64)(59,62)(60,61);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,17,16,18)(8,22,23,12)(9,25,24,29)(13,28,27,30)(21,32,31,26)(33,37,39,38)(34,43,45,44)(35,47,49,48)(36,52,54,53)(40,42,57,56)(41,59,60,50)(46,63,51,64)(55,61,62,58)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258103);