Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471157 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16765 |
| 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 | |
614 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)(3,8,16,28)(4,18)(5,22,23,51)(6,11)(7,12)(9,20)(10,24,34,38)(13,35)(14,26,40,30)(15,42)(17,29,37,57)(19,47,48,50)(21,36)(25,32)(27,55)(31,52,33,49)(39,60)(41,54,46,58)(43,62)(44,63)(45,56)(53,64)(59,61);(1,3,13,37,6,16,7,17)(2,8,12,29,11,28,35,57)(4,14,25,46,20,40,21,41)(5,15,38,55,23,44,24,45)(9,30,32,58,18,26,36,54)(10,27,22,42,34,56,51,63)(19,39,52,64,48,43,49,61)(31,59,50,62,33,53,47,60);(1,4,6,20)(2,9,11,18)(3,14,16,40)(5,19,23,48)(7,25,13,21)(8,26,28,30)(10,33,34,31)(12,36,35,32)(15,43,44,39)(17,46,37,41)(22,50,51,47)(24,52,38,49)(27,53,56,59)(29,58,57,54)(42,62,63,60)(45,61,55,64);(1,5,6,23)(2,10)(3,15,16,44)(4,19,20,48)(7,24,13,38)(8,27)(9,31)(11,34)(12,22)(14,39,40,43)(17,45,37,55)(18,33)(21,49,25,52)(26,53)(28,56)(29,42)(30,59)(32,50)(35,51)(36,47)(41,61,46,64)(54,60)(57,63)(58,62)Polycyclic presentation:
2^9|a^2=efg,b^2=efh,c^2=i,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,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, 378354);