Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #474230 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16913 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
609 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,15,20,33,18,9,19,34)(4,21,25,10,26,37,24,38)(5,11,29,41,28,40,30,42)(16,35,22,43,46,58,47,31)(17,32,48,59,49,44,50,60)(23,51,53,39,27,55,54,36)(45,62,56,63,64,61,52,57);(1,3,6,19,8,18,7,20)(2,9,13,34,14,15,12,33)(4,22,24,16,26,47,25,46)(5,17,29,48,28,49,30,50)(10,35,37,43,38,58,21,31)(11,32,42,60,40,44,41,59)(23,52,53,45,27,56,54,64)(36,61,55,63,39,62,51,57);(1,4,8,26)(2,10,14,38)(3,16,18,46)(5,27,28,23)(6,25,7,24)(9,31,15,43)(11,39,40,36)(12,37,13,21)(17,45,49,64)(19,22,20,47)(29,53,30,54)(32,57,44,63)(33,35,34,58)(41,51,42,55)(48,52,50,56)(59,61,60,62);(1,5,6,29,8,28,7,30)(2,11,12,41,14,40,13,42)(3,17,19,48,18,49,20,50)(4,23,24,53,26,27,25,54)(9,32,33,59,15,44,34,60)(10,36,21,51,38,39,37,55)(16,45,47,56,46,64,22,52)(31,57,58,62,43,63,35,61)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=i,d^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,h^b=hi,d^c=de,f^c=fi,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, 381427);