Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #467050 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16557 |
| 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 | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,8,13,7,12)(3,15,18,9,19,42,31,34)(4,20,23,37,25,10,24,38)(5,26,29,11,30,40,28,41)(16,35,21,32,46,43,51,57)(17,33,49,59,50,60,48,44)(22,39,54,52,55,62,27,36)(45,56,53,58,64,63,47,61);(1,3,6,18,8,19,7,31)(2,9,12,15,13,34,14,42)(4,21,23,46,25,51,24,16)(5,17,29,49,30,50,28,48)(10,35,37,57,20,43,38,32)(11,33,26,44,41,60,40,59)(22,53,54,64,55,47,27,45)(36,61,62,63,52,58,39,56);(1,4,8,25)(2,10,13,20)(3,16,19,46)(5,27,30,54)(6,24,7,23)(9,32,34,57)(11,39,41,62)(12,38,14,37)(15,43,42,35)(17,47,50,53)(18,51,31,21)(22,29,55,28)(26,52,40,36)(33,58,60,61)(44,63,59,56)(45,49,64,48);(1,5)(2,11)(3,17)(4,22)(6,29)(7,28)(8,30)(9,33)(10,36)(12,26)(13,41)(14,40)(15,44)(16,45)(18,49)(19,50)(20,52)(21,53)(23,54)(24,27)(25,55)(31,48)(32,56)(34,60)(35,61)(37,62)(38,39)(42,59)(43,58)(46,64)(47,51)(57,63)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,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, 374247);