Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #433187 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15474 |
| 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 | |
907 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)(3,13,17,8)(4,18)(5,23)(6,26)(7,11)(9,21)(10,25)(12,28)(14,39,19,47)(15,35,43,30)(16,31,45,38)(20,49)(22,37)(24,36)(27,34)(29,41,32,44)(33,54)(40,60,48,64)(42,57,63,59)(46,51)(50,52)(53,55)(56,61,58,62);(1,3,6,16,7,17,28,45)(2,8,26,38,11,13,12,31)(4,19,21,44,22,14,27,41)(5,15,24,42,25,43,55,63)(9,32,37,47,34,29,18,39)(10,35,53,57,23,30,36,59)(20,48,50,62,51,40,54,61)(33,56,49,60,52,58,46,64);(1,4,7,22)(2,9,11,34)(3,14,17,19)(5,20,25,51)(6,27,28,21)(8,29,13,32)(10,33,23,52)(12,37,26,18)(15,40,43,48)(16,44,45,41)(24,54,55,50)(30,56,35,58)(31,39,38,47)(36,49,53,46)(42,62,63,61)(57,60,59,64);(1,5)(2,10,11,23)(3,15)(4,20)(6,24)(7,25)(8,30,13,35)(9,33,34,52)(12,36,26,53)(14,40)(16,42)(17,43)(18,46,37,49)(19,48)(21,50)(22,51)(27,54)(28,55)(29,56,32,58)(31,57,38,59)(39,60,47,64)(41,61)(44,62)(45,63)Polycyclic presentation:
2^9|a^2=g,b^2=ef,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340384);