Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455952 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16272 |
| 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 | |
1109 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)(5,24)(6,14)(7,12)(8,13)(9,18)(10,22)(11,25)(16,41)(17,46)(19,32)(21,34)(23,38)(26,39)(27,37)(28,35)(29,36)(30,45)(31,49)(33,44)(40,50)(42,58)(43,51)(47,59)(48,56)(52,54)(53,57)(55,60)(61,62)(63,64);(1,3,6,27,8,18,7,19)(2,9,12,37,13,15,14,32)(4,16,22,45,23,43,28,44)(5,17,25,50,26,48,29,49)(10,33,20,51,35,30,38,41)(11,31,24,56,36,40,39,46)(21,42,53,63,55,47,54,62)(34,59,52,64,60,58,57,61);(1,4)(2,10)(3,16,18,43)(5,21)(6,28)(7,22)(8,23)(9,30,15,33)(11,34)(12,38)(13,35)(14,20)(17,47,48,42)(19,45,27,44)(24,57)(25,54)(26,55)(29,53)(31,59,40,58)(32,41,37,51)(36,60)(39,52)(46,64,56,61)(49,62,50,63);(1,5,8,26)(2,11,13,36)(3,17,18,48)(4,21,23,55)(6,25,7,29)(9,31,15,40)(10,34,35,60)(12,24,14,39)(16,42,43,47)(19,49,27,50)(20,52,38,57)(22,53,28,54)(30,58,33,59)(32,46,37,56)(41,61,51,64)(44,62,45,63)Polycyclic presentation:
2^9|b^2=ef,c^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,f^c=fi,g^c=gi,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, 363149);