Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437585 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2403 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5921 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)(4,20)(5,11)(6,13)(7,12)(8,14)(9,19)(10,23)(16,29)(17,42)(18,39)(21,50)(22,35)(24,34)(25,40)(26,38)(27,31)(28,37)(30,48)(32,44)(33,51)(36,47)(41,45)(43,54)(46,56)(49,55)(52,59)(53,58)(57,61)(60,63)(62,64);(1,3)(2,9)(4,16)(5,17)(6,27)(7,19)(8,18)(10,32)(11,36)(12,39)(13,15)(14,31)(20,41)(21,43)(22,49)(23,45)(24,44)(25,48)(26,46)(28,47)(29,35)(30,38)(33,54)(34,55)(37,56)(40,42)(50,64)(51,63)(52,61)(53,62)(57,58)(59,60);(1,4,8,24)(2,10,14,35)(3,16,18,44)(5,21,26,52)(6,23,7,22)(9,29,31,32)(11,33,38,58)(12,20,13,34)(15,41,39,55)(17,43,46,61)(19,49,27,45)(25,53,28,51)(30,54,36,57)(37,59,40,50)(42,60,56,64)(47,63,48,62);(1,5)(2,11)(3,17)(4,21)(6,28)(7,25)(8,26)(9,30)(10,33)(12,40)(13,37)(14,38)(15,42)(16,43)(18,46)(19,48)(20,50)(22,53)(23,51)(24,52)(27,47)(29,54)(31,36)(32,57)(34,59)(35,58)(39,56)(41,60)(44,61)(45,63)(49,62)(55,64);(3,18)(9,31)(15,39)(16,44)(17,46)(19,27)(29,32)(30,36)(41,55)(42,56)(43,61)(45,49)(47,48)(54,57)(60,64)(62,63)Polycyclic presentation:
2^9|c^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,f^c=fi,g^c=gi,f^d=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72303362, #72303730, #72303748, … |
| maximal normal subgroup of | #72303362, #72303730, #72303748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344782);