Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13291 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1993 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1993 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
757 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)(3,14,20,42)(4,21,25,53)(5,11)(6,27)(7,30)(8,13)(9,18,34,50)(10,23,37,26)(12,28)(15,38)(16,45)(17,40,19,41)(22,54)(24,57,58,36)(29,59)(31,56)(32,43,33,49)(35,47)(39,46)(44,61,55,52)(48,62)(51,63)(60,64);(1,3,15,43,5,17,29,50,8,20,28,49,7,19,6,18)(2,9,27,40,30,33,12,14,13,34,59,41,11,32,38,42)(4,22,55,64,23,16,24,56,25,46,44,47,26,48,58,63)(10,35,61,39,53,31,57,45,37,60,52,54,21,51,36,62);(1,4)(2,10,13,37)(3,16,20,48)(5,23)(6,24)(7,26)(8,25)(9,31)(11,21,30,53)(12,36,38,57)(14,39)(15,44)(17,46,19,22)(18,47,50,64)(27,52,59,61)(28,55)(29,58)(32,35)(33,60)(34,51)(40,62)(41,45)(42,54)(43,63,49,56);(1,5,8,7)(2,11,13,30)(3,17,20,19)(4,23,25,26)(6,15,29,28)(9,32,34,33)(10,21,37,53)(12,27,38,59)(14,40,42,41)(16,46,48,22)(18,43,50,49)(24,44,58,55)(31,35,51,60)(36,52,57,61)(39,62,54,45)(47,63,64,56)Polycyclic presentation:
2^9|a^2=g,b^2=efg,c^2=fh,d^2=i,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ch,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249773, #72249778, #72249832, … |
| maximal normal subgroup of | #72249773, #72249778, #72249832, … |
| maximal quotient of | #72256839, C2×#495392, #72322955, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402588);