Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455956 |
| 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 | |
1102 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)(4,10)(5,25)(6,14)(7,13)(8,12)(9,18)(11,28)(16,35)(17,32)(19,34)(20,51)(21,52)(22,38)(23,37)(24,39)(26,55)(27,42)(29,41)(30,33)(31,45)(36,54)(40,53)(43,48)(44,57)(46,61)(47,62)(49,59)(50,60)(56,64)(58,63);(1,3,8,30,7,19,6,18)(2,9,14,34,13,33,12,15)(4,20,22,16,24,45,23,46)(5,17,27,48,29,50,28,49)(10,35,37,51,39,61,38,31)(11,32,41,59,42,60,25,43)(21,47,54,44,53,63,26,56)(36,62,52,64,55,58,40,57);(1,4,6,22,7,24,8,23)(2,10,12,37,13,39,14,38)(3,16,18,45,19,46,30,20)(5,26,27,53,29,54,28,21)(9,31,15,35,33,51,34,61)(11,40,41,55,42,52,25,36)(17,47,48,56,50,63,49,44)(32,58,59,64,60,62,43,57);(1,5,7,29)(2,11,13,42)(3,17,19,50)(4,21,24,53)(6,28,8,27)(9,32,33,60)(10,36,39,55)(12,25,14,41)(15,43,34,59)(16,44,46,56)(18,49,30,48)(20,47,45,63)(22,54,23,26)(31,57,51,64)(35,62,61,58)(37,52,38,40)Polycyclic presentation:
2^9|b^2=ef,c^2=e,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363153);