Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510514 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31680 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24280 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
555 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)(3,14,19,9)(4,20)(5,11)(6,27)(7,13)(8,12)(10,22)(15,46,49,62)(16,33,17,34)(18,45,53,35)(21,56,58,54)(23,55)(24,38)(25,37)(26,40)(28,60)(29,61,50,42)(30,39)(31,41)(32,48,44,51)(36,57,63,52)(43,59)(47,64);(1,3,6,17,8,19,5,16)(2,9,27,33,12,14,11,34)(4,21,22,57,25,58,24,52)(7,29,28,53,31,50,26,18)(10,36,37,56,38,63,20,54)(13,42,60,45,41,61,40,35)(15,47,48,23,49,30,51,59)(32,55,46,39,44,43,62,64);(1,4)(2,10)(3,15)(5,22)(6,24)(7,30)(8,25)(9,32)(11,37)(12,38)(13,43)(14,44)(16,48)(17,51)(18,52)(19,49)(20,27)(21,50)(23,28)(26,59)(29,58)(31,47)(33,62)(34,46)(35,54)(36,61)(39,60)(40,64)(41,55)(42,63)(45,56)(53,57);(1,5,8,6)(2,11,12,27)(3,16,19,17)(4,22,25,24)(7,26,31,28)(9,33,14,34)(10,37,38,20)(13,40,41,60)(15,48,49,51)(18,50,53,29)(21,57,58,52)(23,30,59,47)(32,62,44,46)(35,42,45,61)(36,54,63,56)(39,43,64,55)Polycyclic presentation:
2^9|a^2=efh,b^2=ef,d^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=hi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72320520, #72320582, #72320745 |
| maximal normal subgroup of | #72320520, #72320582, #72320745 |
| maximal quotient of | #72321379, #72321585, #72321720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417712);