Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13074 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1893 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1893 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,11,8,13,7,30)(3,14,16,41,18,43,19,51)(4,20,23,36,24,10,21,38)(6,26,25,32,29,56,28,12)(9,31,34,49,35,50,42,17)(15,40,47,64,46,61,44,63)(22,54,55,39,27,57,53,37)(33,59,62,45,52,58,60,48);(1,3,6,17,5,16,25,31,8,18,29,49,7,19,28,50)(2,9,32,43,30,42,26,41,13,35,12,14,11,34,56,51)(4,15,27,48,21,44,55,58,24,46,22,45,23,47,53,59)(10,33,37,61,38,62,54,63,20,52,39,40,36,60,57,64);(1,4)(2,10)(3,15)(5,21)(6,27)(7,23)(8,24)(9,33)(11,36)(12,39)(13,20)(14,40)(16,44)(17,48)(18,46)(19,47)(22,29)(25,55)(26,54)(28,53)(30,38)(31,58)(32,37)(34,60)(35,52)(41,63)(42,62)(43,61)(45,49)(50,59)(51,64)(56,57);(1,5,8,7)(2,11,13,30)(3,16,18,19)(4,21,24,23)(6,25,29,28)(9,34,35,42)(10,36,20,38)(12,26,32,56)(14,41,43,51)(15,44,46,47)(17,31,49,50)(22,53,27,55)(33,60,52,62)(37,57,39,54)(40,63,61,64)(45,59,48,58)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,f^b=fi,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249764, #72249778, #72249825, … |
| maximal normal subgroup of | #72249764, #72249778, #72249825, … |
| maximal quotient of | #72250254, C2×#494666, #72322887 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401862);