Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471081 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16747 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
616 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)(3,14)(4,10)(5,24,27,55)(6,29)(7,12)(8,13)(9,17)(11,28,40,26)(15,44)(16,32,49,59)(18,34)(19,42)(20,39,52,62)(21,41)(22,37)(23,38)(25,36,53,61)(30,33)(31,46)(35,50)(43,48,54,56)(45,57,51,60)(47,63,64,58);(1,3,7,17,8,18,6,30)(2,9,12,34,13,33,29,14)(4,19,22,50,23,15,21,46)(5,16,28,48,27,49,26,56)(10,35,37,44,38,31,41,42)(11,32,24,54,40,59,55,43)(20,51,53,64,52,45,25,47)(36,60,39,58,61,57,62,63);(1,4,8,23)(2,10,13,38)(3,15,18,19)(5,25,27,53)(6,22,7,21)(9,31,33,35)(11,39,40,62)(12,41,29,37)(14,42,34,44)(16,47,49,64)(17,50,30,46)(20,28,52,26)(24,36,55,61)(32,58,59,63)(43,57,54,60)(45,56,51,48);(1,5,6,26,8,27,7,28)(2,11,12,24,13,40,29,55)(3,16,30,56,18,49,17,48)(4,20,21,25,23,52,22,53)(9,32,34,54,33,59,14,43)(10,36,37,39,38,61,41,62)(15,45,50,64,19,51,46,47)(31,57,42,63,35,60,44,58)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=i,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,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, 378278);