Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #384334 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14377 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
974 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)(3,13,17,8)(4,18)(5,23,25,36)(6,26)(7,11)(9,21)(10,24,35,56)(12,28)(14,39,19,47)(15,42)(16,31,27,37)(20,32,51,59)(22,46)(29,41,38,49)(30,43)(33,52)(34,55)(40,58)(44,53)(45,50,54,64)(48,57)(60,63)(61,62);(1,3)(2,8)(4,19)(5,15)(6,27)(7,17)(9,29)(10,34)(11,13)(12,37)(14,22)(16,28)(18,39)(20,48)(21,41)(23,53)(24,55)(25,44)(26,31)(30,35)(32,58)(33,38)(36,42)(40,51)(43,56)(45,62)(46,47)(49,52)(50,61)(54,60)(57,59)(63,64);(1,4,6,21,7,22,28,52)(2,9,12,18,11,33,26,46)(3,14,16,41,17,19,27,49)(5,20,24,50,25,51,56,64)(8,29,31,47,13,38,37,39)(10,32,36,45,35,59,23,54)(15,40,43,61,44,48,55,63)(30,57,42,62,34,58,53,60);(1,5,7,25)(2,10)(3,15,17,44)(4,20,22,51)(6,24,28,56)(8,30)(9,32)(11,35)(12,36)(13,34)(14,40,19,48)(16,43,27,55)(18,45)(21,50,52,64)(23,26)(29,57)(31,42)(33,59)(37,53)(38,58)(39,60)(41,61,49,63)(46,54)(47,62)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291531);