Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498941 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2021 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2083 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,13,17,35)(4,18,19,38)(5,22,26,10)(6,27,29,11)(8,23,32,15)(9,28,33,21)(14,41,42,63)(16,46,48,40)(20,51,24,50)(25,37,57,54)(30,59,60,64)(31,45,58,55)(34,53,36,56)(39,47,62,44)(43,49,52,61);(1,3)(2,8)(4,14)(5,23)(6,16)(7,17)(9,30)(10,35)(11,31)(12,32)(13,22)(15,26)(18,49)(19,42)(20,52)(21,44)(24,43)(25,55)(27,58)(28,47)(29,48)(33,60)(34,62)(36,39)(37,46)(38,61)(40,54)(41,51)(45,57)(50,63)(53,59)(56,64);(1,4)(2,9)(3,14)(5,24)(6,28)(7,19)(8,30)(10,36)(11,38)(12,33)(13,39)(15,43)(16,47)(17,42)(18,27)(20,26)(21,29)(22,34)(23,52)(25,53)(31,61)(32,60)(35,62)(37,50)(40,63)(41,46)(44,48)(45,59)(49,58)(51,54)(55,64)(56,57);(4,19)(9,33)(14,42)(18,38)(20,24)(21,28)(30,60)(34,36)(39,62)(41,63)(43,52)(44,47)(49,61)(50,51)(53,56)(59,64)Polycyclic presentation:
2^9|a^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=ghi,c^b=cg,e^b=ei,h^b=hi,d^c=di,e^c=eh,f^c=fi,g^e=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72280539, #72281946, #72282134, … |
| maximal normal subgroup of | #72280539, #72281946, #72282134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406138);