Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424591 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15323 |
| 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 | |
1176 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)(3,14,18,39)(4,10)(5,25)(6,12)(7,30)(8,13)(9,17,33,19)(11,28)(15,43,20,41)(16,32,47,59)(21,50)(22,35)(23,36)(24,55)(26,37,48,42)(27,38)(29,40)(31,46,58,45)(34,54)(44,64,49,57)(51,60,56,63)(52,61)(53,62);(1,3)(2,9)(4,20)(5,26)(6,19)(7,17)(8,18)(10,31)(11,37)(12,39)(13,33)(14,30)(15,23)(16,27)(21,51)(22,45)(24,46)(25,32)(28,47)(29,48)(34,60)(35,43)(36,58)(38,42)(40,59)(41,55)(44,54)(49,52)(50,57)(53,56)(61,63)(62,64);(1,4)(2,10)(3,15,18,20)(5,21)(6,22)(7,24)(8,23)(9,31,33,58)(11,34)(12,35)(13,36)(14,41,39,43)(16,44,47,49)(17,45,19,46)(25,50)(26,56,48,51)(27,52)(28,54)(29,53)(30,55)(32,57,59,64)(37,60,42,63)(38,61)(40,62);(1,5,8,29)(2,11,13,38)(3,16,18,47)(4,21,23,53)(6,28,7,27)(9,32,33,59)(10,34,36,61)(12,40,30,25)(14,42,39,37)(15,44,20,49)(17,48,19,26)(22,54,24,52)(31,57,58,64)(35,62,55,50)(41,63,43,60)(45,51,46,56)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,e^d=ei,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331788);