Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7487870 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #52004 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1830 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)(4,9)(5,10)(6,11)(7,13)(8,17)(12,26)(15,43)(16,42)(18,29)(19,32)(20,34)(21,40)(22,28)(23,37)(24,39)(25,30)(27,52)(31,46)(33,51)(35,47)(36,49)(38,54)(41,45)(44,60)(48,56)(50,61)(53,57)(55,63)(58,62)(59,64);(1,3)(2,8)(4,15)(5,22)(6,25)(7,18)(9,31)(10,35)(11,14)(12,29)(13,30)(16,23)(17,26)(19,48)(20,45)(21,52)(24,53)(27,34)(28,37)(32,58)(33,41)(36,60)(38,57)(39,42)(40,43)(44,50)(46,51)(47,54)(49,63)(55,59)(56,61)(62,64);(1,4)(2,9)(3,15)(5,19)(6,21)(7,20)(8,27)(10,36)(11,40)(12,33)(13,34)(14,41)(16,44)(17,46)(18,45)(22,48)(23,50)(24,49)(25,52)(26,51)(28,56)(29,43)(30,31)(32,39)(35,58)(37,59)(38,61)(42,60)(47,62)(53,63)(54,64)(55,57);(1,5,6,23,7,24,26,54)(2,10,11,37,13,39,12,38)(3,16,17,22,18,47,25,53)(4,19,21,50,20,49,51,64)(8,28,29,35,30,57,14,42)(9,32,40,61,34,36,33,59)(15,44,46,48,45,62,52,63)(27,55,43,60,31,56,41,58);(2,11,13,12)(4,20)(8,29,30,14)(9,33,34,40)(10,37,39,38)(15,45)(19,49)(21,51)(27,41,31,43)(28,35,57,42)(32,59,36,61)(44,62)(46,52)(48,63)(50,64)(55,58,56,60)Polycyclic presentation:
2^9|d^2=fh,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ef,f^a=fi,h^a=hi,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=ei,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72284483, #72317825 |
| maximal normal subgroup of | #72284483, #72317825 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7395066);