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 | |
3643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 64 points:
(1,2)(3,8)(4,9)(5,10)(6,11)(7,12)(13,14)(15,16)(17,18,23,20)(19,26,22,24)(21,28,30,25)(27,31,29,32)(33,34,39,40)(35,43,44,41)(36,48,51,37)(38,53,54,42)(45,55,49,46)(47,56,58,57)(50,52,63,61)(59,60,62,64);(1,2,7,12)(3,8,4,9)(5,6,11,10)(13,16,15,14)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(34,40)(35,44)(37,48)(42,53)(45,49)(47,58)(52,61)(59,62);(1,3,7,4)(2,8,12,9)(5,13,11,15)(6,16,10,14)(17,20,23,18)(19,24,22,26)(21,28,30,25)(27,31,29,32)(33,35,39,44)(34,41,40,43)(36,49,51,45)(37,46,48,55)(38,47,54,58)(42,56,53,57)(50,62,63,59)(52,60,61,64);(1,4,7,3)(2,9,12,8)(5,13,11,15)(6,16,10,14)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,36)(34,37)(35,45)(38,50)(39,51)(40,48)(41,55)(42,52)(43,46)(44,49)(47,59)(53,61)(54,63)(56,64)(57,60)(58,62);(1,5,7,11)(2,10,12,6)(3,13,4,15)(8,14,9,16)(17,19,23,22)(18,26,20,24)(21,27,30,29)(25,32,28,31)(33,37)(34,36)(35,46)(38,52)(39,48)(40,51)(41,49)(42,50)(43,45)(44,55)(47,60)(53,63)(54,61)(56,62)(57,59)(58,64);(1,6,7,10)(2,11,12,5)(3,14,4,16)(8,13,9,15)(17,22,23,19)(18,24,20,26)(21,29,30,27)(25,31,28,32)(33,38,39,54)(34,42,40,53)(35,47,44,58)(36,50,51,63)(37,52,48,61)(41,56,43,57)(45,59,49,62)(46,60,55,64)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=g,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9388732);