Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453856 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16158 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,29)(3,14,17,9,19,34,18,35)(4,20,23,10,24,40,30,37)(5,25,27,39,28,11,31,56)(15,42,45,41,21,32,44,59)(16,38,46,60,47,54,26,33)(22,50,57,36,53,48,52,55)(43,63,64,61,49,58,51,62);(1,3,6,18,7,19,8,17)(2,9,29,14,13,35,12,34)(4,21,23,44,24,15,30,45)(5,26,27,16,28,46,31,47)(10,32,40,59,37,42,20,41)(11,38,56,60,25,54,39,33)(22,51,52,49,53,64,57,43)(36,61,50,63,55,62,48,58);(1,4,7,24)(2,10,13,37)(3,15)(5,22,28,53)(6,30,8,23)(9,32)(11,36,25,55)(12,40,29,20)(14,41)(16,43)(17,45)(18,44)(19,21)(26,51)(27,57,31,52)(33,58)(34,59)(35,42)(38,62)(39,50,56,48)(46,64)(47,49)(54,61)(60,63);(1,5)(2,11)(3,16,19,47)(4,22)(6,27)(7,28)(8,31)(9,33,35,60)(10,36)(12,39)(13,25)(14,38,34,54)(15,43,21,49)(17,26,18,46)(20,48)(23,52)(24,53)(29,56)(30,57)(32,58,42,63)(37,55)(40,50)(41,62,59,61)(44,64,45,51)Polycyclic presentation:
2^9|a^2=eg,b^2=eh,c^2=ef,d^2=efi,e^2=i,f^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,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361053);