Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(2,6)(7,12)(8,13)(14,16)(17,18,23,28)(19,27,22,24)(20,30,29,25)(21,26,31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,53)(44,54)(45,51)(46,48)(47,55)(49,56)(50,52)(57,59)(58,63)(60,62)(61,64);(4,11)(8,13)(9,15)(14,16)(17,19,23,22)(18,24,28,27)(20,31,29,21)(25,32,30,26)(34,39)(40,51)(41,46)(42,52)(53,57)(54,56)(55,60)(63,64);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,20)(18,25)(19,21)(22,31)(23,29)(24,26)(27,32)(28,30)(33,35,38,45)(34,40,39,51)(36,43,48,59)(37,49,50,44)(41,53,46,57)(42,56,52,54)(47,61,62,58)(55,64,60,63);(1,3,5,10)(2,7,6,12)(4,9,11,15)(8,14,13,16)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,30)(31,32)(33,36,38,48)(34,41,39,46)(35,43,45,59)(37,47,50,62)(40,53,51,57)(42,55,52,60)(44,58,49,61)(54,63,56,64);(1,4,5,11)(2,8,6,13)(3,9,10,15)(7,14,12,16)(17,21,23,31)(18,26,28,32)(19,29,22,20)(24,30,27,25)(33,34)(35,40)(36,46)(37,42)(38,39)(41,48)(43,57)(44,54)(45,51)(47,60)(49,56)(50,52)(53,59)(55,62)(58,64)(61,63);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,22,23,19)(18,27,28,24)(20,31,29,21)(25,32,30,26)(33,37,38,50)(34,42,39,52)(35,44,45,49)(36,47,48,62)(40,54,51,56)(41,55,46,60)(43,58,59,61)(53,63,57,64)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833910);