Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1898 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1898 |
| prime radical | #60366 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#585 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,5,11)(3,15,18,44,21,46,20,45)(4,22,23,10,26,38,25,40)(6,27,16,42,30,12,31,33)(9,32,36,28,37,53,35,19)(17,48,51,43,52,63,49,61)(24,58,47,39,29,56,57,41)(34,54,60,64,62,50,55,59);(1,3,16,32)(2,9,33,44)(4,17,47,59)(5,18,30,19)(6,28,7,20)(8,21,31,53)(10,34,56,48)(11,35,27,46)(12,15,13,36)(14,37,42,45)(22,55,41,43)(23,49,24,54)(25,51,29,50)(26,52,57,64)(38,60,39,61)(40,62,58,63);(1,4)(2,10)(3,17)(5,23)(6,29)(7,25)(8,26)(9,34)(11,38)(12,41)(13,22)(14,40)(15,43)(16,47)(18,49)(19,54)(20,51)(21,52)(24,30)(27,39)(28,50)(31,57)(32,59)(33,56)(35,60)(36,55)(37,62)(42,58)(44,48)(45,63)(46,61)(53,64);(1,5,8,7)(2,11,14,13)(3,18,21,20)(4,23,26,25)(6,16,30,31)(9,35,37,36)(10,38,40,22)(12,33,27,42)(15,44,46,45)(17,49,52,51)(19,53,28,32)(24,57,29,47)(34,60,62,55)(39,58,41,56)(43,48,61,63)(50,59,54,64)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280647, #72281882, #72282105, … |
| maximal normal subgroup of | #72280647, #72281882, #72282105, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401953);