Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417665 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,3,4)(5,6,10,17,12,18,11,16)(7,19,22,13,24,38,23,39)(8,25,29,43,30,42,28,14)(9,31,33,15,35,45,34,46)(20,49,51,40,52,36,26,47)(21,37,32,48,54,61,53,44)(27,57,59,55,60,41,58,62)(50,65,56,66,68,67,63,64);(5,7,10,22,12,24,11,23)(6,13,16,19,18,39,17,38)(8,26,28,52,30,51,29,20)(9,32,33,54,35,53,34,21)(14,40,25,36,43,47,42,49)(15,44,31,61,46,48,45,37)(27,50,58,63,60,68,59,56)(41,66,62,67,57,64,55,65);(2,4)(5,8,12,30)(6,14,18,43)(7,20,24,52)(9,27,35,60)(10,28,11,29)(13,36,39,49)(15,41,46,57)(16,25,17,42)(19,47,38,40)(21,50,54,68)(22,26,23,51)(31,62,45,55)(32,63,53,56)(33,58,34,59)(37,64,61,66)(44,65,48,67);(5,9,12,35)(6,15,18,46)(7,21,24,54)(8,27,30,60)(10,34,11,33)(13,37,39,61)(14,41,43,57)(16,45,17,31)(19,48,38,44)(20,50,52,68)(22,53,23,32)(25,55,42,62)(26,56,51,63)(28,59,29,58)(36,64,49,66)(40,65,47,67)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326211);