Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359679 |
| 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 | C2×#3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,14)(3,15,18,9,19,34,30,33)(4,20,23,37,25,10,24,38)(5,26,27,11,29,40,28,39)(16,42,21,31,45,35,44,57)(17,32,47,58,48,59,46,41)(22,51,53,36,54,55,52,49)(43,62,50,61,63,56,64,60);(1,3)(2,9)(4,21)(5,17)(6,30)(7,18)(8,19)(10,35)(11,32)(12,34)(13,33)(14,15)(16,24)(20,42)(22,50)(23,45)(25,44)(26,41)(27,47)(28,46)(29,48)(31,37)(36,60)(38,57)(39,59)(40,58)(43,52)(49,61)(51,56)(53,63)(54,64)(55,62);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,31)(11,36)(12,37)(13,20)(14,38)(15,35)(17,43)(18,44)(19,45)(21,30)(26,55)(27,52)(28,53)(29,54)(32,56)(33,57)(34,42)(39,49)(40,51)(41,60)(46,50)(47,64)(48,63)(58,61)(59,62);(1,5,8,29)(2,11,13,39)(3,17,19,48)(4,22,25,54)(6,28,7,27)(9,32,33,59)(10,36,20,49)(12,40,14,26)(15,41,34,58)(16,43,45,63)(18,47,30,46)(21,50,44,64)(23,53,24,52)(31,56,57,62)(35,60,42,61)(37,51,38,55)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^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,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72287270, #72287292, #72287345, … |
| maximal normal subgroup of | #72287270, #72287292, #72287345, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266878);