Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13867 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3707 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)(4,18,20,36)(5,22,24,10)(6,12)(7,11)(9,21,33,25)(14,40)(15,29)(16,39,48,30)(19,49)(23,37,26,34)(27,35)(28,43)(31,46)(32,51)(38,45)(41,60,61,64)(42,55)(44,56)(47,57)(50,59)(52,53)(54,62,58,63);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,31)(10,29)(11,13)(12,30)(18,40)(19,41)(20,42)(21,43)(22,38)(23,44)(24,45)(25,46)(26,47)(27,48)(28,33)(32,58)(34,56)(35,39)(36,55)(37,57)(49,60)(50,61)(51,62)(52,63)(53,54)(59,64);(1,4)(2,9)(3,14)(5,19)(6,25)(7,20)(8,28)(10,32)(11,33)(12,36)(13,31)(15,41)(16,46)(17,42)(18,35)(21,27)(22,53)(23,52)(24,50)(26,51)(29,54)(30,40)(34,49)(37,59)(38,58)(39,55)(43,48)(44,63)(45,61)(47,62)(56,64)(57,60);(1,5,7,24)(2,10,11,22)(3,15,17,45)(4,19,20,50)(6,26,27,23)(8,29,13,38)(9,32,33,53)(12,37,35,34)(14,41,42,61)(16,47,48,44)(18,49,36,59)(21,52,25,51)(28,54,31,58)(30,57,39,56)(40,60,55,64)(43,63,46,62)Polycyclic presentation:
2^9|a^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72302933, #72303070, #72303071, … |
| maximal normal subgroup of | #72302933, #72303070, #72303071, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266731);