Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #404013 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14885 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2500 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)(3,15)(4,10)(5,24)(6,14)(7,12)(8,13)(9,18)(11,27)(16,40)(17,45)(19,39)(20,49)(21,36)(22,34)(23,35)(25,38)(26,37)(28,31)(29,48)(30,47)(32,43)(33,52)(41,46)(42,55)(44,56)(50,58)(51,53)(54,59)(57,62)(60,61)(63,64);(1,3)(2,9)(4,16)(5,17)(6,28)(7,18)(8,19)(10,32)(11,30)(12,39)(13,31)(14,15)(20,42)(21,48)(22,43)(23,44)(24,41)(25,54)(26,47)(27,46)(29,35)(33,57)(34,56)(36,40)(37,59)(38,45)(49,60)(50,63)(51,62)(52,61)(53,64)(55,58);(1,4,8,23)(2,10,13,35)(3,16,19,44)(5,20,26,51)(6,22,7,21)(9,29,31,32)(11,33,38,58)(12,36,14,34)(15,40,39,56)(17,42,47,62)(18,48,28,43)(24,53,37,49)(25,52,27,50)(30,55,45,57)(41,60,59,64)(46,63,54,61);(1,5)(2,11,13,38)(3,17)(4,20)(6,25)(7,27)(8,26)(9,30,31,45)(10,33,35,58)(12,37,14,24)(15,41,39,59)(16,42)(18,46)(19,47)(21,50)(22,52)(23,51)(28,54)(29,55,32,57)(34,53,36,49)(40,60,56,64)(43,61)(44,62)(48,63)Polycyclic presentation:
2^9|c^2=i,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,e^c=ei,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72251694, #72251718, #72251731, … |
| maximal normal subgroup of | #72251694, #72251718, #72251731, … |
| maximal quotient of | #72255920 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311210);