Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6364250 |
| 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 | #62098 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1489 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,17,23)(4,9,20,27)(5,11,6,10)(8,15,24,16)(14,29,30,25)(18,32,31,22)(19,28,21,26)(33,34,39,43)(35,40,46,55)(36,48,50,56)(37,42,38,51)(41,52,57,49)(44,58,60,63)(45,54,47,61)(53,62,64,59);(1,3,7,17)(2,8,12,24)(4,18,20,31)(5,16,6,15)(9,25,27,29)(10,13,11,23)(14,21,30,19)(22,28,32,26)(33,35)(34,40)(36,44)(37,45)(38,47)(39,46)(41,53)(42,54)(43,55)(48,58)(49,59)(50,60)(51,61)(52,62)(56,63)(57,64);(33,36)(34,41)(35,44)(37,49)(38,52)(39,50)(40,53)(42,56)(43,57)(45,59)(46,60)(47,62)(48,51)(54,63)(55,64)(58,61);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,26)(11,28)(12,27)(13,29)(15,18)(16,31)(17,30)(23,25)(24,32)(33,37,39,38)(34,42,43,51)(35,45,46,47)(36,49,50,52)(40,54,55,61)(41,56,57,48)(44,59,60,62)(53,63,64,58);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,23,24,13)(9,26,27,28)(14,18,30,31)(22,25,32,29)(35,46)(40,55)(44,60)(45,47)(53,64)(54,61)(58,63)(59,62)Polycyclic presentation:
2^9|a^2=hi,b^2=h,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6271625);