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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1087 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)(3,15)(4,21)(5,28)(6,12)(7,33)(8,14)(9,18)(10,26)(11,29)(13,32)(16,46)(17,43)(19,38)(20,37)(22,34)(23,39)(24,54)(25,42)(27,41)(30,44)(31,59)(35,50)(36,49)(40,57)(45,48)(47,64)(51,52)(53,56)(55,62)(58,63)(60,61);(1,3,8,20)(2,9)(4,22)(5,17,30,50)(6,31,29,49)(7,18,32,19)(10,34,42,46)(11,35)(12,43)(13,15)(14,38)(16,27)(21,52,41,45)(23,55)(24,58)(25,51)(26,48)(28,36)(33,37)(39,61,40,63)(44,59)(47,57)(53,64,54,62)(56,60);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,27)(9,34)(11,39)(12,40)(13,41)(14,42)(15,45)(17,47)(18,48)(19,51)(20,22)(21,33)(26,32)(28,54)(29,56)(30,57)(31,60)(35,61)(36,62)(37,52)(38,46)(43,63)(44,53)(49,58)(50,55)(59,64);(1,5,7,29,8,30,32,6)(2,11,13,28,14,12,33,44)(3,17,18,49,20,50,19,31)(4,23,25,56,27,57,26,24)(9,35,15,36,38,43,37,59)(10,39,41,54,42,40,21,53)(16,47,48,58,22,55,51,60)(34,61,45,62,46,63,52,64);(1,6,32,30,8,29,7,5)(2,12,13,44,14,11,33,28)(3,17,18,49,20,50,19,31)(4,24,26,57,27,56,25,23)(9,36,37,35,38,59,15,43)(10,40,41,53,42,39,21,54)(16,47,48,58,22,55,51,60)(34,62,52,61,46,64,45,63)Polycyclic presentation:
2^9|b^2=fgh,d^2=f,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, 6711046);