Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479323 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17199 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
628 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)(3,15,19,35)(4,20)(5,26)(6,12)(7,13)(8,14)(9,18,36,32)(10,23)(11,31)(16,45,47,33)(17,44,27,56)(21,37,50,60)(22,41)(24,40)(25,39)(28,58)(29,43)(30,42)(34,49,55,57)(38,53)(46,63,64,59)(48,62,52,61)(51,54);(1,3,8,19)(2,9)(4,21,25,50)(5,27,31,17)(6,18,7,32)(10,37)(11,34)(12,15)(13,35)(14,36)(16,23,47,24)(20,45)(22,52,54,48)(26,55)(28,46,53,64)(29,57,30,49)(33,39)(38,62)(40,60)(41,59)(42,44)(43,56)(51,63)(58,61);(1,4,8,25)(2,10,14,40)(3,16,19,47)(5,28,31,53)(6,24,7,23)(9,33,36,45)(11,41,26,51)(12,20,13,39)(15,37,35,60)(17,48,27,52)(18,50,32,21)(22,30,54,29)(34,61,55,62)(38,43,58,42)(44,59,56,63)(46,57,64,49);(1,5,6,29,8,31,7,30)(2,11,13,43,14,26,12,42)(3,17,18,49,19,27,32,57)(4,22,23,28,25,54,24,53)(9,34,35,56,36,55,15,44)(10,38,20,51,40,58,39,41)(16,46,21,48,47,64,50,52)(33,59,37,62,45,63,60,61)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=i,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386520);