Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #400510 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14802 |
| 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 | |
977 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,30)(3,14,18,33,19,40,17,9)(4,20,24,35,25,37,23,10)(5,26,28,58,29,11,27,36)(15,31,43,48,21,50,42,38)(16,44,46,32,47,39,45,56)(22,52,54,34,55,49,53,57)(41,61,63,60,51,59,62,64);(1,3,6,17,8,19,7,18)(2,9,12,14,13,33,30,40)(4,21,23,43,25,15,24,42)(5,16,27,45,29,47,28,46)(10,31,20,48,35,50,37,38)(11,32,36,39,26,56,58,44)(22,51,53,63,55,41,54,62)(34,59,49,64,57,61,52,60);(1,4,8,25)(2,10,13,35)(3,15,19,21)(5,22,29,55)(6,24,7,23)(9,31,33,50)(11,34,26,57)(12,37,30,20)(14,38,40,48)(16,41,47,51)(17,43,18,42)(27,54,28,53)(32,59,56,61)(36,52,58,49)(39,60,44,64)(45,63,46,62);(1,5)(2,11,13,26)(3,16)(4,22)(6,27)(7,28)(8,29)(9,32,33,56)(10,34,35,57)(12,36,30,58)(14,39,40,44)(15,41)(17,45)(18,46)(19,47)(20,49,37,52)(21,51)(23,53)(24,54)(25,55)(31,59,50,61)(38,60,48,64)(42,62)(43,63)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,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, 307707);