Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2381 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14670 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,15,20,9,19,34,18,33)(4,10,23,37,24,38,30,44)(5,11,28,42,29,43,27,41)(16,35,21,53,48,45,52,31)(17,46,25,32,51,39,50,60)(22,36,26,40,54,61,57,64)(47,59,49,62,55,63,56,58);(1,3)(2,9)(4,21)(5,25,29,50)(6,20)(7,18)(8,19)(10,35)(11,39,43,46)(12,34)(13,15)(14,33)(16,23)(17,27,51,28)(22,47,54,55)(24,52)(26,56,57,49)(30,48)(31,37)(32,41,60,42)(36,58,61,62)(38,45)(40,63,64,59)(44,53);(1,4)(2,10)(3,16,19,48)(5,26)(6,30)(7,23)(8,24)(9,31,33,53)(11,40)(12,44)(13,37)(14,38)(15,45,34,35)(17,49,51,56)(18,52,20,21)(22,27)(25,55,50,47)(28,54)(29,57)(32,59,60,63)(36,41)(39,62,46,58)(42,61)(43,64);(1,5,6,27,8,29,7,28)(2,11,12,41,14,43,13,42)(3,17,18,50,19,51,20,25)(4,22,23,26,24,54,30,57)(9,32,15,46,33,60,34,39)(10,36,37,40,38,61,44,64)(16,47,21,49,48,55,52,56)(31,58,35,59,53,62,45,63)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299196);