Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462393 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16429 |
| 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 | |
715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,7,13,6,12)(3,15,29,9,19,34,18,33)(4,20,22,37,23,10,30,36)(5,24,26,39,28,40,27,11)(16,43,45,31,46,41,56,51)(17,32,48,60,50,61,49,42)(21,53,55,52,54,38,25,35)(44,63,57,59,64,58,47,62);(1,3,7,19)(2,9,13,33)(4,16,23,46)(5,17,28,50)(6,29,8,18)(10,31,20,51)(11,32,39,61)(12,34,14,15)(21,44,54,64)(22,56,30,45)(24,42,40,60)(25,57,55,47)(26,49,27,48)(35,58,52,63)(36,43,37,41)(38,62,53,59);(1,4,6,22,7,23,8,30)(2,10,12,36,13,20,14,37)(3,16,18,45,19,46,29,56)(5,25,26,54,28,55,27,21)(9,31,15,41,33,51,34,43)(11,38,24,52,39,53,40,35)(17,47,48,64,50,57,49,44)(32,59,60,63,61,62,42,58);(1,5)(2,11,13,39)(3,17)(4,21)(6,27)(7,28)(8,26)(9,32,33,61)(10,35,20,52)(12,40,14,24)(15,42,34,60)(16,44)(18,49)(19,50)(22,55)(23,54)(25,30)(29,48)(31,58,51,63)(36,53,37,38)(41,62,43,59)(45,57)(46,64)(47,56)Polycyclic presentation:
2^9|a^2=eg,b^2=i,c^2=e,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,g^c=gi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369590);