Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #403988 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14885 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2494 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)(5,25)(6,13)(7,12)(8,14)(9,18)(10,23)(11,28)(16,30)(17,45)(19,39)(21,50)(22,36)(24,35)(26,37)(27,38)(29,32)(31,56)(33,44)(34,51)(40,43)(41,47)(42,59)(46,54)(48,58)(49,53)(52,55)(57,62)(60,64)(61,63);(1,3)(2,9)(4,16)(5,17)(6,29)(7,18)(8,19)(10,33)(11,31)(12,39)(13,15)(14,32)(20,40)(21,42)(22,48)(23,43)(24,44)(25,54)(26,56)(27,46)(28,47)(30,36)(34,59)(35,58)(37,45)(38,41)(49,60)(50,63)(51,64)(52,61)(53,62)(55,57);(1,4,8,24)(2,10,14,36)(3,16,19,44)(5,21,28,53)(6,23,7,22)(9,30,32,33)(11,34,25,55)(12,20,13,35)(15,40,39,58)(17,42,47,62)(18,48,29,43)(26,52,27,51)(31,57,54,59)(37,49,38,50)(41,60,45,63)(46,64,56,61);(1,5)(2,11,14,25)(3,17)(4,21)(6,26)(7,27)(8,28)(9,31,32,54)(10,34,36,55)(12,37,13,38)(15,41,39,45)(16,42)(18,46)(19,47)(20,49,35,50)(22,51)(23,52)(24,53)(29,56)(30,57,33,59)(40,60,58,63)(43,61)(44,62)(48,64)Polycyclic presentation:
2^9|c^2=i,d^2=g,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,e^c=ei,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311185);