Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #28182 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13993 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
915 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,29)(3,14,19,32,18,41,17,9)(4,10,23,35,24,37,22,36)(5,25,27,38,28,54,26,11)(15,33,20,30,44,39,43,56)(16,45,48,61,47,31,46,40)(21,50,52,59,53,58,51,34)(42,55,49,60,63,64,62,57);(1,3)(2,9,13,32)(4,20,24,43)(5,16)(6,17)(7,19)(8,18)(10,33)(11,31,38,45)(12,14,29,41)(15,23,44,22)(21,49,53,62)(25,40,54,61)(26,46)(27,48)(28,47)(30,35)(34,57)(36,56)(37,39)(42,52,63,51)(50,55)(58,64)(59,60);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,24)(9,30)(11,34)(12,35)(13,37)(14,39)(16,42)(17,43)(18,44)(19,20)(25,50)(26,51)(27,52)(28,53)(29,36)(31,55)(32,56)(33,41)(38,59)(40,60)(45,64)(46,62)(47,63)(48,49)(54,58)(57,61);(1,5,6,26,8,28,7,27)(2,11,12,25,13,38,29,54)(3,16,17,46,18,47,19,48)(4,21,22,51,24,53,23,52)(9,31,14,40,32,45,41,61)(10,34,35,50,37,59,36,58)(15,42,43,62,44,63,20,49)(30,55,39,60,56,64,33,57)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273645);