Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449419 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15943 |
| 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 | |
678 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)(3,15)(4,10)(5,25,28,42)(6,13)(7,12)(8,14)(9,18)(11,29,41,27)(16,35)(17,32,46,53)(19,34)(20,33)(21,48)(22,50,52,54)(23,38)(24,39)(26,55,47,40)(30,37)(31,44)(36,51,62,57)(43,61,63,58)(45,59)(49,64,56,60);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,21,23,16,24,44,30,45)(5,26,29,17,28,47,27,46)(10,35,37,48,39,59,38,31)(11,40,42,32,41,55,25,53)(22,43,57,49,52,63,51,56)(36,61,50,60,62,58,54,64);(1,4,6,23,8,24,7,30)(2,10,13,38,14,39,12,37)(3,16,19,21,20,45,18,44)(5,22,27,51,28,52,29,57)(9,31,15,35,34,48,33,59)(11,36,42,50,41,62,25,54)(17,43,47,49,46,63,26,56)(32,58,40,60,53,61,55,64);(1,5,8,28)(2,11)(3,17)(4,22,24,52)(6,27,7,29)(9,32,34,53)(10,36)(12,25)(13,42)(14,41)(15,40,33,55)(16,43)(18,26)(19,47)(20,46)(21,49)(23,51,30,57)(31,58,48,61)(35,60,59,64)(37,54)(38,50)(39,62)(44,56)(45,63)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=g,d^2=eg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356615);