Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #470944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16720 |
| 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 | |
669 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)(3,15)(4,20,23,39)(5,26)(6,12)(7,13)(8,14)(9,18)(10,24,38,25)(11,29)(16,45,47,36)(17,48)(19,35)(21,32,51,43)(22,37,27,40)(28,42)(30,41)(31,34)(33,52)(44,50)(46,60,64,63)(49,57)(53,55,58,56)(54,61,59,62);(1,3)(2,9,14,15)(4,21,23,51)(5,17)(6,31)(7,19)(8,18)(10,36)(11,33,42,44)(12,35,13,34)(16,25,47,24)(20,32)(22,54,27,59)(26,57,41,48)(28,52)(29,50)(30,49)(37,61)(38,45)(39,43)(40,62)(46,55,64,56)(53,60)(58,63);(1,4,6,25,8,23,7,24)(2,10,12,39,14,38,13,20)(3,16,31,21,18,47,19,51)(5,27,28,55,30,22,29,56)(9,32,35,36,15,43,34,45)(11,40,26,58,42,37,41,53)(17,46,52,59,49,64,50,54)(33,60,57,62,44,63,48,61);(1,5,8,30)(2,11,14,42)(3,17,18,49)(4,22,23,27)(6,29,7,28)(9,33,15,44)(10,37,38,40)(12,41,13,26)(16,46,47,64)(19,52,31,50)(20,53,39,58)(21,54,51,59)(24,56,25,55)(32,60,43,63)(34,57,35,48)(36,61,45,62)Polycyclic presentation:
2^9|a^2=efg,b^2=efh,c^2=ef,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 378141);