Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6803848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #39148 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3067 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1075 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,13)(3,14,19,35)(4,20,25,39)(5,26,28,55)(6,11,30,41)(7,31,27,12)(9,18,36,17)(10,23,40,24)(15,37,46,32)(16,43,48,33)(21,38,52,62)(22,50,51,54)(29,34,47,58)(42,49,59,45)(44,61,63,57)(53,60,64,56);(1,3)(2,9,13,36)(4,15,25,46)(5,16)(6,29)(7,17)(8,19)(10,37)(11,33,41,43)(12,14,31,35)(18,27)(20,42)(21,44,52,63)(22,53,51,64)(23,45,24,49)(26,34,55,58)(28,48)(30,47)(32,40)(38,60)(39,59)(50,57)(54,61)(56,62);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,25)(9,32)(11,38)(12,39)(13,40)(14,42)(16,44)(17,45)(18,49)(19,46)(20,31)(24,27)(26,54)(28,52)(29,53)(30,51)(33,56)(34,57)(35,59)(36,37)(41,62)(43,60)(47,64)(48,63)(50,55)(58,61);(1,5,27,30,8,28,7,6)(2,11,31,55,13,41,12,26)(3,16,18,47,19,48,17,29)(4,21,24,51,25,52,23,22)(9,33,35,58,36,43,14,34)(10,38,20,50,40,62,39,54)(15,44,49,64,46,63,45,53)(32,56,59,61,37,60,42,57);(1,6,7,28,8,30,27,5)(2,11,31,55,13,41,12,26)(3,16,18,47,19,48,17,29)(4,22,23,52,25,51,24,21)(9,34,14,43,36,58,35,33)(10,38,20,50,40,62,39,54)(15,44,49,64,46,63,45,53)(32,57,42,60,37,61,59,56)Polycyclic presentation:
2^9|a^2=i,b^2=fgh,d^2=fi,e^2=fhi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,f^a=fi,g^a=gi,h^a=hi,c^b=ch,e^b=efh,g^b=gi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6711048);