Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471821 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16807 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,14)(3,15,18,9,19,33,27,32)(4,10,28,35,22,36,21,37)(5,23,29,11,26,39,25,40)(16,43,45,30,46,41,51,60)(17,47,49,56,50,31,52,42)(20,53,55,34,54,38,24,57)(44,63,48,61,64,59,58,62);(1,3,8,18,7,19,6,27)(2,9,12,33,13,32,14,15)(4,16,28,45,22,46,21,51)(5,17,25,52,26,50,29,49)(10,30,35,41,36,60,37,43)(11,31,23,56,40,47,39,42)(20,44,24,58,54,64,55,48)(34,59,53,61,57,63,38,62);(1,4,7,22)(2,10,13,36)(3,16,19,46)(5,24,26,55)(6,28,8,21)(9,30,32,60)(11,38,40,53)(12,37,14,35)(15,41,33,43)(17,48,50,58)(18,51,27,45)(20,29,54,25)(23,57,39,34)(31,61,47,62)(42,63,56,59)(44,52,64,49);(1,5)(2,11)(3,17)(4,20)(6,29)(7,26)(8,25)(9,31)(10,34)(12,23)(13,40)(14,39)(15,42)(16,44)(18,52)(19,50)(21,55)(22,54)(24,28)(27,49)(30,59)(32,47)(33,56)(35,53)(36,57)(37,38)(41,61)(43,62)(45,58)(46,64)(48,51)(60,63)Polycyclic presentation:
2^9|a^2=ef,b^2=gh,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379018);