Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493621 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1790 |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62886 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2310 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)(3,13,17,35)(4,18,19,38)(5,22,26,10)(6,11)(7,12)(8,15)(9,21,33,27)(14,43)(16,42,51,61)(20,54)(23,32)(24,53)(25,39,28,36)(29,40)(30,58,59,64)(31,48)(34,56)(37,50)(41,49)(44,63)(45,52,47,60)(46,62)(55,57);(1,3,7,17)(2,8,12,32)(4,14,19,44)(5,23,26,15)(6,16,29,51)(9,30,33,59)(10,35,22,13)(11,37,40,31)(18,52,38,60)(20,47,24,45)(21,46,27,49)(25,50,28,48)(34,62,57,41)(36,42,39,61)(43,53,63,54)(55,58,56,64);(1,4)(2,9)(3,14)(5,24)(6,27)(7,19)(8,30)(10,34)(11,38)(12,33)(13,41)(15,47)(16,49)(17,44)(18,40)(20,26)(21,29)(22,57)(23,45)(25,55)(28,56)(31,52)(32,59)(35,62)(36,53)(37,60)(39,54)(42,63)(43,61)(46,51)(48,64)(50,58);(4,19)(9,33)(14,44)(18,38)(20,24)(21,27)(30,59)(34,57)(41,62)(43,63)(45,47)(46,49)(52,60)(53,54)(55,56)(58,64)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=ehi,f^c=fi,g^c=gi,f^e=fi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72303715, #72303722, #72303738, … |
| maximal normal subgroup of | #72303715, #72303722, #72303738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400818);