Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460622 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16368 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
947 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)(3,8,15,28)(4,9)(5,20,21,38)(6,12)(7,11)(10,22,35,24)(13,30,40,26)(14,42)(16,29,46,56)(17,49,50,60)(18,33)(19,32)(23,37)(25,36)(27,48)(31,51,34,52)(39,61)(41,57,47,54)(43,64)(44,58)(45,55)(53,62)(59,63);(1,3)(2,8,11,28)(4,13)(5,14)(6,16)(7,15)(9,30,32,26)(10,27,35,55)(12,29,36,56)(17,39)(18,41)(19,40)(20,42,38,58)(21,44)(22,45)(23,47)(24,48)(25,46)(31,59,34,53)(33,57,37,54)(43,50)(49,61,60,64)(51,62)(52,63);(1,4,7,19)(2,9,11,32)(3,13,15,40)(5,17,21,50)(6,23,25,18)(8,26,28,30)(10,34,35,31)(12,37,36,33)(14,43,44,39)(16,47,46,41)(20,49,38,60)(22,52,24,51)(27,53,55,59)(29,57,56,54)(42,61,58,64)(45,62,48,63);(1,5)(2,10)(3,14)(4,17)(6,24)(7,21)(8,27)(9,31)(11,35)(12,38)(13,39)(15,44)(16,48)(18,52)(19,50)(20,36)(22,25)(23,51)(26,53)(28,55)(29,58)(30,59)(32,34)(33,60)(37,49)(40,43)(41,63)(42,56)(45,46)(47,62)(54,61)(57,64)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72285868, #72286106, #72286134, … |
| maximal normal subgroup of | #72285868, #72286106, #72286134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367819);