Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471284 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16786 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
652 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)(3,8,16,27)(4,9,19,30)(5,21,22,49)(6,11)(7,12)(10,23,33,37)(13,34)(14,25)(15,41)(17,28,36,56)(18,46)(20,31,24,35)(26,54)(29,50)(32,48)(38,60,42,62)(39,53)(40,52)(43,63)(44,55)(45,57)(47,59)(51,61,58,64);(1,3,13,36,6,16,7,17)(2,8,12,28,11,27,34,56)(4,14,24,45,19,39,20,40)(5,15,37,54,22,43,23,44)(9,25,31,52,30,53,35,57)(10,26,21,41,33,55,49,63)(18,38,50,64,47,42,48,61)(29,51,46,60,32,58,59,62);(1,4,6,19)(2,9,11,30)(3,14)(5,18,22,47)(7,24,13,20)(8,25)(10,32,33,29)(12,35,34,31)(15,42)(16,39)(17,45)(21,46,49,59)(23,50,37,48)(26,51)(27,53)(28,57)(36,40)(38,43)(41,62)(44,61)(52,56)(54,64)(55,58)(60,63);(1,5)(2,10,11,33)(3,15)(4,18)(6,22)(7,23)(8,26,27,55)(9,29,30,32)(12,21,34,49)(13,37)(14,38)(16,43)(17,44)(19,47)(20,48)(24,50)(25,51,53,58)(28,41,56,63)(31,46,35,59)(36,54)(39,42)(40,61)(45,64)(52,60,57,62)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=ef,d^2=fi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378481);