Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454037 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16249 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2129 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)(3,14)(4,10)(5,11)(6,29)(7,12)(8,13)(9,17)(15,33)(16,48)(18,40)(19,32)(20,53)(21,34)(22,41)(23,35)(24,36)(25,56)(26,42)(27,39)(28,38)(30,46)(31,50)(37,55)(43,47)(44,52)(45,58)(49,57)(51,59)(54,61)(60,64)(62,63);(1,3)(2,9)(4,20)(5,16)(6,19)(7,17)(8,18)(10,33)(11,31)(12,40)(13,32)(14,29)(15,23)(21,49)(22,47)(24,46)(25,45)(26,51)(27,52)(28,50)(30,35)(34,60)(36,43)(37,62)(38,44)(39,59)(41,53)(42,48)(54,63)(55,64)(56,57)(58,61);(1,4,8,24)(2,10,13,36)(3,15,18,47)(5,25,27,55)(6,23,7,22)(9,30,32,53)(11,37,39,56)(12,41,29,35)(14,43,40,33)(16,49,52,63)(17,20,19,46)(21,28,54,26)(31,58,59,60)(34,42,61,38)(44,62,48,57)(45,51,64,50);(1,5)(2,11)(3,16)(4,21)(6,28)(7,26)(8,27)(9,31)(10,34)(12,42)(13,39)(14,44)(15,45)(17,51)(18,52)(19,50)(20,49)(22,55)(23,25)(24,54)(29,38)(30,57)(32,59)(33,60)(35,56)(36,61)(37,41)(40,48)(43,58)(46,63)(47,64)(53,62)Polycyclic presentation:
2^9|c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72264928, #72264941, #72265017, … |
| maximal normal subgroup of | #72264928, #72264941, #72265017, … |
| maximal quotient of | #72265741 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361234);