Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437593 |
| 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 | |
5873 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)(4,20)(5,11)(6,29)(7,13)(8,12)(9,17)(10,24)(15,45)(16,44)(18,33)(19,34)(21,43)(22,51)(23,37)(25,38)(26,32)(27,42)(28,40)(30,41)(31,47)(35,50)(36,54)(39,49)(46,62)(48,57)(52,61)(53,59)(55,60)(56,64)(58,63);(1,3,8,19)(2,9,12,33)(4,21,25,50)(5,26,28,48)(6,18,7,17)(10,35,37,43)(11,39,40,44)(13,34,29,14)(15,24,47,23)(16,30,49,27)(20,31,38,45)(22,46,55,64)(32,41,57,42)(36,56,59,62)(51,61,60,58)(52,53,63,54);(1,4)(2,10)(3,15)(5,22)(6,23)(7,24)(8,25)(9,31)(11,36)(12,37)(13,38)(14,43)(16,46)(17,21)(18,50)(19,47)(20,29)(26,52)(27,53)(28,55)(30,54)(32,56)(33,45)(34,35)(39,58)(40,59)(41,60)(42,51)(44,61)(48,63)(49,64)(57,62);(1,5)(2,11)(3,16)(4,22)(6,27)(7,30)(8,28)(9,32)(10,36)(12,40)(13,41)(14,44)(15,46)(17,26)(18,48)(19,49)(20,51)(21,52)(23,53)(24,54)(25,55)(29,42)(31,56)(33,57)(34,39)(35,58)(37,59)(38,60)(43,61)(45,62)(47,64)(50,63);(2,12)(9,33)(10,37)(11,40)(13,29)(14,34)(20,38)(31,45)(32,57)(35,43)(36,59)(39,44)(41,42)(51,60)(56,62)(58,61)Polycyclic presentation:
2^9|b^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,g^c=gi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72286994, #72287270, #72287286, … |
| maximal normal subgroup of | #72286994, #72287270, #72287286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344790);