Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #474022 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16893 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
871 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)(3,9,16,33)(4,19)(5,24,26,40)(6,27)(7,13)(8,12)(10,22)(11,25,39,29)(14,41,20,51)(15,44)(17,48,18,32)(21,38,54,35)(23,37)(28,36)(30,49,34,43)(31,50)(42,59)(45,56)(46,57)(47,58)(52,55,60,53)(61,62)(63,64);(1,3,7,18,8,16,6,17)(2,9,27,48,12,33,13,32)(4,20,22,49,23,14,28,43)(5,15,29,50,26,46,25,47)(10,34,19,51,36,30,37,41)(11,31,24,44,39,58,40,57)(21,45,53,63,54,42,55,62)(35,59,52,64,38,56,60,61);(1,4,6,22,8,23,7,28)(2,10,13,19,12,36,27,37)(3,14,17,43,16,20,18,49)(5,21,25,53,26,54,29,55)(9,30,32,41,33,34,48,51)(11,38,40,60,39,35,24,52)(15,45,47,63,46,42,50,62)(31,56,57,61,58,59,44,64);(1,5)(2,11)(3,15)(4,21)(6,25)(7,29)(8,26)(9,31)(10,35)(12,39)(13,40)(14,42)(16,46)(17,47)(18,50)(19,52)(20,45)(22,53)(23,54)(24,27)(28,55)(30,56)(32,57)(33,58)(34,59)(36,38)(37,60)(41,61)(43,62)(44,48)(49,63)(51,64)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,f^2=i,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,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381218);