Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418535 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
916 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,12,18)(7,19,22,13)(8,24,29,40)(9,30,31,43)(10,17,11,16)(14,28,41,27)(15,32,42,33)(20,46,25,55)(21,49,51,63)(23,37,34,44)(26,57,58,39)(35,48,38,52)(36,50,45,53)(47,64,56,62)(54,59,61,60)(65,67,68,66);(5,7,10,23,12,22,11,34)(6,13,16,37,18,19,17,44)(8,25,27,52,29,20,28,48)(9,21,32,50,31,51,33,53)(14,38,24,46,41,35,40,55)(15,36,43,63,42,45,30,49)(26,56,59,67,58,47,60,66)(39,64,61,68,57,62,54,65);(5,8,12,29)(6,14,18,41)(7,20,22,25)(9,26,31,58)(10,28,11,27)(13,35,19,38)(15,39,42,57)(16,40,17,24)(21,47,51,56)(23,52,34,48)(30,61,43,54)(32,60,33,59)(36,62,45,64)(37,46,44,55)(49,65,63,68)(50,67,53,66);(2,4)(5,9)(6,15)(7,21,22,51)(8,26)(10,33)(11,32)(12,31)(13,36,19,45)(14,39)(16,30)(17,43)(18,42)(20,47,25,56)(23,53,34,50)(24,54)(27,60)(28,59)(29,58)(35,62,38,64)(37,49,44,63)(40,61)(41,57)(46,65,55,68)(48,67,52,66)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 355845);