Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393986 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3878 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)(4,20)(5,11)(6,13)(7,12)(8,14)(9,18)(10,23)(16,29)(17,45)(19,38)(21,50)(22,35)(24,34)(25,39)(26,37)(27,31)(28,36)(30,46)(32,44)(33,51)(40,43)(41,48)(42,57)(47,56)(49,55)(52,59)(53,58)(54,63)(60,61)(62,64);(1,3)(2,9)(4,16)(5,17)(6,27)(7,18)(8,19)(10,32)(11,30)(12,38)(13,15)(14,31)(20,40)(21,42)(22,49)(23,43)(24,44)(25,47)(26,48)(28,46)(29,35)(33,57)(34,55)(36,41)(37,56)(39,45)(50,60)(51,62)(52,63)(53,61)(54,58)(59,64);(1,4,8,24)(2,10,14,35)(3,16,19,44)(5,21,26,52)(6,23,7,22)(9,29,31,32)(11,33,37,58)(12,20,13,34)(15,40,38,55)(17,42,48,63)(18,49,27,43)(25,53,28,51)(30,54,56,57)(36,59,39,50)(41,60,45,64)(46,62,47,61);(1,5)(2,11)(3,17)(4,21)(6,28)(7,25)(8,26)(9,30)(10,33)(12,39)(13,36)(14,37)(15,41)(16,42)(18,47)(19,48)(20,50)(22,53)(23,51)(24,52)(27,46)(29,54)(31,56)(32,57)(34,59)(35,58)(38,45)(40,60)(43,62)(44,63)(49,61)(55,64)Polycyclic presentation:
2^9|c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,f^c=fi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72280729, #72282605, #72282641, … |
| maximal normal subgroup of | #72280729, #72282605, #72282641, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301183);