Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427436 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15369 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,13,7,14)(3,15,18,33,20,34,19,9)(4,21,23,40,24,10,29,37)(5,25,27,39,28,41,30,11)(16,44,46,35,47,42,49,31)(17,32,26,38,48,60,50,43)(22,52,54,51,55,57,56,36)(45,61,53,62,64,59,58,63);(1,3,8,20)(2,9,13,33)(4,16,24,47)(5,26,28,50)(6,19,7,18)(10,35,21,31)(11,38,39,43)(12,34,14,15)(17,30,48,27)(22,53,55,58)(23,49,29,46)(25,32,41,60)(36,62,51,63)(37,44,40,42)(45,56,64,54)(52,61,57,59);(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,24)(9,31)(11,36)(12,40)(13,21)(14,37)(15,42)(17,45)(18,49)(19,46)(20,47)(25,57)(26,58)(27,56)(28,55)(30,54)(32,59)(33,35)(34,44)(38,62)(39,51)(41,52)(43,63)(48,64)(50,53)(60,61);(1,5)(2,11,13,39)(3,17)(4,22)(6,30)(7,27)(8,28)(9,32,33,60)(10,36,21,51)(12,41,14,25)(15,43,34,38)(16,45)(18,50)(19,26)(20,48)(23,56)(24,55)(29,54)(31,59,35,61)(37,57,40,52)(42,63,44,62)(46,58)(47,64)(49,53)Polycyclic presentation:
2^9|a^2=e,b^2=i,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^c=ei,e^d=ei,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334633);