Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23892 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14127 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2110 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,14,6,13)(3,15,18,9,20,33,19,34)(4,21,25,10,26,37,24,38)(5,27,28,11,30,39,29,40)(16,42,44,35,45,31,22,50)(17,32,47,57,48,58,46,41)(23,52,53,49,54,36,55,60)(43,59,62,56,63,64,51,61);(1,3)(2,9)(4,22)(5,17)(6,19)(7,18)(8,20)(10,35)(11,32)(12,33)(13,15)(14,34)(16,25)(21,42)(23,51)(24,45)(26,44)(27,41)(28,47)(29,46)(30,48)(31,37)(36,59)(38,50)(39,57)(40,58)(43,53)(49,61)(52,64)(54,62)(55,63)(56,60);(1,4,7,25,8,26,6,24)(2,10,12,37,14,38,13,21)(3,16,18,44,20,45,19,22)(5,23,28,53,30,54,29,55)(9,31,33,50,34,42,15,35)(11,36,39,60,40,52,27,49)(17,43,47,62,48,63,46,51)(32,56,57,64,58,61,41,59);(1,5)(2,11)(3,17)(4,23)(6,29)(7,28)(8,30)(9,32)(10,36)(12,39)(13,27)(14,40)(15,41)(16,43)(18,47)(19,46)(20,48)(21,49)(22,51)(24,55)(25,53)(26,54)(31,56)(33,57)(34,58)(35,59)(37,60)(38,52)(42,61)(44,62)(45,63)(50,64)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280857);