Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419899 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15273 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1527 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)(4,10)(5,11,26,40)(6,28)(7,12)(8,13)(9,17)(15,42)(16,45,46,38)(18,32)(19,41)(20,30)(21,50,53,34)(22,36)(23,37)(24,54,47,31)(25,55,27,39)(29,35)(33,44)(43,62,51,59)(48,58)(49,57,63,64)(52,60,56,61);(1,3)(2,9)(4,20)(5,24)(6,18)(7,17)(8,19)(10,33)(11,38)(12,41)(13,32)(14,28)(15,22)(16,25)(21,51)(23,48)(26,47)(27,46)(29,44)(30,36)(31,39)(34,57)(35,58)(37,42)(40,45)(43,53)(49,52)(50,64)(54,55)(56,63)(59,60)(61,62);(1,4,8,23)(2,10,13,37)(3,15,19,44)(5,21,26,53)(6,22,7,29)(9,30,32,58)(11,34,40,50)(12,35,28,36)(14,42,41,33)(16,43,46,51)(17,20,18,48)(24,49,47,63)(25,52,27,56)(31,57,54,64)(38,62,45,59)(39,60,55,61);(1,5,6,25,8,26,7,27)(2,11,12,39,13,40,28,55)(3,16,17,24,19,46,18,47)(4,21,22,52,23,53,29,56)(9,31,14,38,32,54,41,45)(10,34,35,60,37,50,36,61)(15,43,20,49,44,51,48,63)(30,57,42,62,58,64,33,59)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327096);