Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465630 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16526 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1039 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)(4,19)(5,25)(6,30)(7,12)(8,13)(10,22)(11,27)(14,35)(15,32)(16,48)(17,33)(18,34)(20,49)(21,52)(23,37)(24,40)(26,50)(28,39)(29,56)(31,42)(36,54)(38,53)(41,62)(43,61)(44,63)(45,55)(46,64)(47,60)(51,59)(57,58);(1,3,7,17,8,18,6,16)(2,9,12,33,13,34,30,48)(4,20,23,43,24,42,22,14)(5,15,29,47,27,45,28,46)(10,35,19,49,37,61,40,31)(11,32,39,60,25,55,56,64)(21,51,54,44,26,57,53,41)(36,62,50,59,38,63,52,58);(1,4,8,24)(2,10,13,37)(3,14,18,43)(5,26,27,21)(6,23,7,22)(9,31,34,49)(11,38,25,36)(12,40,30,19)(15,44,45,41)(16,20,17,42)(28,53,29,54)(32,59,55,58)(33,61,48,35)(39,50,56,52)(46,57,47,51)(60,62,64,63);(1,5,6,28,8,27,7,29)(2,11,12,39,13,25,30,56)(3,15,16,46,18,45,17,47)(4,21,22,53,24,26,23,54)(9,32,33,60,34,55,48,64)(10,36,19,50,37,38,40,52)(14,41,42,57,43,44,20,51)(31,58,35,62,49,59,61,63)Polycyclic presentation:
2^9|b^2=g,c^2=i,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372827);