Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #382440 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2309 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14338 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,28)(3,9,16,32,17,34,29,33)(4,18,22,37,21,38,27,10)(5,11,24,40,26,41,25,55)(14,35,44,48,43,62,19,30)(15,31,46,59,47,61,56,60)(20,49,51,39,23,52,54,36)(42,63,53,58,45,64,50,57);(1,3)(2,9)(4,19)(5,15)(6,16)(7,29)(8,17)(10,35)(11,31)(12,32)(13,34)(14,22)(18,48)(20,50)(21,44)(23,53)(24,46)(25,56)(26,47)(27,43)(28,33)(30,38)(36,63)(37,62)(39,64)(40,59)(41,61)(42,51)(45,54)(49,58)(52,57)(55,60);(1,4)(2,10)(3,14,17,43)(5,23,26,20)(6,27)(7,22)(8,21)(9,30,34,48)(11,39,41,36)(12,38)(13,37)(15,45)(16,19,29,44)(18,28)(24,51,25,54)(31,58)(32,62,33,35)(40,49,55,52)(42,47)(46,53)(50,56)(57,61)(59,63)(60,64);(1,5)(2,11)(3,15)(4,20)(6,24)(7,25)(8,26)(9,31)(10,36)(12,40)(13,41)(14,42)(16,46)(17,47)(18,49)(19,50)(21,23)(22,51)(27,54)(28,55)(29,56)(30,57)(32,59)(33,60)(34,61)(35,63)(37,39)(38,52)(43,45)(44,53)(48,58)(62,64)Polycyclic presentation:
2^9|a^2=f,c^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285753, #72285784, #72286021, … |
| maximal normal subgroup of | #72285753, #72285784, #72286021, … |
| maximal quotient of | C2×#382570 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289766);