Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442596 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#535 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2535 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2281 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)(3,14)(4,10)(5,11)(6,28)(7,13)(8,12)(9,17)(15,42)(16,38)(18,41)(19,32)(20,30)(21,34)(22,37)(23,35)(24,52)(25,40)(26,53)(27,39)(29,36)(31,46)(33,44)(43,63)(45,62)(47,56)(48,55)(49,57)(50,61)(51,60)(54,59)(58,64);(1,3)(2,9)(4,20)(5,24)(6,19)(7,17)(8,18)(10,33)(11,38)(12,32)(13,41)(14,28)(15,22)(16,27)(21,49)(23,47)(25,46)(26,45)(29,44)(30,37)(31,39)(34,58)(35,42)(36,56)(40,62)(43,50)(48,51)(52,53)(54,64)(55,61)(57,60)(59,63);(1,4)(2,10)(3,15,18,44)(5,21)(6,29)(7,22)(8,23)(9,30,32,56)(11,34)(12,35)(13,37)(14,33,41,42)(16,43,45,49)(17,47,19,20)(24,48,46,64)(25,50)(26,54)(27,51)(28,36)(31,55,52,58)(38,57,62,63)(39,60)(40,61)(53,59);(1,5,7,27,8,25,6,26)(2,11,13,39,12,40,28,53)(3,16,19,24,18,45,17,46)(4,21,22,51,23,50,29,54)(9,31,14,38,32,52,41,62)(10,34,37,60,35,61,36,59)(15,43,20,48,44,49,47,64)(30,55,33,57,56,58,42,63);(2,12)(5,25)(9,32)(10,35)(13,28)(14,41)(16,45)(21,50)(24,46)(26,27)(30,56)(33,42)(36,37)(43,49)(48,64)(51,54)Polycyclic presentation:
2^9|c^2=g,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,h^b=hi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370103);