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 | #60387 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2077 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)(3,14)(4,20)(5,11)(6,26)(7,31)(8,13)(9,17)(10,23)(12,30)(15,43)(16,42)(18,50)(19,39)(21,36)(22,54)(24,37)(25,57)(27,34)(28,52)(29,38)(32,59)(33,48)(35,53)(40,56)(41,46)(44,62)(45,51)(47,63)(49,58)(55,61)(60,64);(1,3)(2,9)(4,15)(5,16)(6,27)(7,18)(8,19)(10,32)(11,33)(12,39)(13,34)(14,26)(17,30)(20,51)(21,44)(22,55)(23,46)(24,47)(25,49)(28,45)(29,48)(31,58)(35,62)(36,60)(37,61)(38,50)(40,41)(42,57)(43,54)(52,63)(53,59)(56,64);(1,4,8,24)(2,10,13,36)(3,15,19,47)(5,23,7,21)(6,28,30,22)(9,32,34,60)(11,37,31,20)(12,40,26,35)(14,41,39,62)(16,46,18,44)(17,45,27,55)(25,53,29,56)(33,61,58,51)(38,52,57,54)(42,63,50,43)(48,59,49,64);(1,5,8,7)(2,11,13,31)(3,16,19,18)(4,21,24,23)(6,25,30,29)(9,33,34,58)(10,20,36,37)(12,38,26,57)(14,42,39,50)(15,44,47,46)(17,48,27,49)(22,53,28,56)(32,51,60,61)(35,52,40,54)(41,43,62,63)(45,64,55,59)Polycyclic presentation:
2^9|c^2=i,d^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, #72281917, #72282045, … |
| maximal normal subgroup of | #72280539, #72281917, #72282045, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406140);