Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456739 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16285 |
| 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:
(1,2,6,26,7,12,30,11)(3,13,27,33,17,40,16,8)(4,18,21,36,22,9,28,41)(5,10,29,38,25,39,24,42)(14,43,51,31,19,54,46,34)(15,47,49,32,50,44,52,60)(20,53,57,62,56,35,23,37)(45,63,48,61,55,59,64,58);(1,3,6,27,7,17,30,16)(2,8,26,13,12,33,11,40)(4,19,21,46,22,14,28,51)(5,15,29,49,25,50,24,52)(9,34,41,43,18,31,36,54)(10,32,38,44,39,60,42,47)(20,55,57,64,56,45,23,48)(35,61,37,59,53,58,62,63);(1,4,7,22)(2,9,12,18)(3,14,17,19)(5,23,25,57)(6,28,30,21)(8,31,33,34)(10,37,39,62)(11,41,26,36)(13,43,40,54)(15,48,50,64)(16,51,27,46)(20,29,56,24)(32,59,60,63)(35,42,53,38)(44,61,47,58)(45,52,55,49);(1,5)(2,10)(3,15)(4,20)(6,29)(7,25)(8,32)(9,35)(11,42)(12,39)(13,44)(14,45)(16,52)(17,50)(18,53)(19,55)(21,57)(22,56)(23,28)(24,30)(26,38)(27,49)(31,58)(33,60)(34,61)(36,62)(37,41)(40,47)(43,59)(46,64)(48,51)(54,63)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=i,e^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,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363936);