Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1889 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1868 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)(4,18,19,39)(5,10)(6,26,29,11)(7,12)(8,15,32,22)(9,21)(14,43)(16,49)(20,54,23,57)(24,40,28,36)(25,37)(27,33)(30,58,59,64)(31,48)(34,56)(38,51)(41,50,62,46)(42,52)(44,63)(45,60)(47,53)(55,61);(1,3,5,22,7,17,25,15)(2,8,10,35,12,32,37,13)(4,14,23,45,19,44,20,47)(6,16,24,51,29,52,28,48)(9,30,34,62,33,59,61,41)(11,38,36,42,26,31,40,49)(18,53,57,43,39,60,54,63)(21,46,56,64,27,50,55,58);(1,4)(2,9)(3,14)(5,23)(6,27)(7,19)(8,30)(10,34)(11,39)(12,33)(13,41)(15,47)(16,50)(17,44)(18,26)(20,25)(21,29)(22,45)(24,55)(28,56)(31,53)(32,59)(35,62)(36,54)(37,61)(38,60)(40,57)(42,63)(43,49)(46,52)(48,64)(51,58);(4,19)(9,33)(14,44)(18,39)(20,23)(21,27)(30,59)(34,61)(41,62)(43,63)(45,47)(46,50)(53,60)(54,57)(55,56)(58,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=i,f^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=eh,f^c=fi,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72303777, #72303781, #72303791, … |
| maximal normal subgroup of | #72303777, #72303781, #72303791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401806);