Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17101 |
| 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 | |
1020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,14,7,12)(3,15,19,35,20,34,18,9)(4,21,24,39,25,38,31,10)(5,26,28,11,30,42,29,43)(16,32,22,52,47,58,46,36)(17,40,50,44,51,33,49,55)(23,37,27,56,53,61,54,41)(45,59,63,62,64,60,48,57);(1,3,6,19,8,20,7,18)(2,9,13,15,14,35,12,34)(4,22,24,47,25,46,31,16)(5,17,28,50,30,51,29,49)(10,36,21,32,39,52,38,58)(11,40,42,44,43,33,26,55)(23,48,27,45,53,63,54,64)(37,60,56,57,61,59,41,62);(1,4)(2,10)(3,16)(5,27)(6,31)(7,24)(8,25)(9,32)(11,41)(12,21)(13,38)(14,39)(15,36)(17,48)(18,22)(19,46)(20,47)(23,28)(26,37)(29,53)(30,54)(33,59)(34,52)(35,58)(40,60)(42,61)(43,56)(44,62)(45,49)(50,64)(51,63)(55,57);(1,5,8,30)(2,11,14,43)(3,17,20,51)(4,23,25,53)(6,29,7,28)(9,33,35,40)(10,37,39,61)(12,42,13,26)(15,44,34,55)(16,45,47,64)(18,50,19,49)(21,41,38,56)(22,48,46,63)(24,54,31,27)(32,57,58,62)(36,59,52,60)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384196);