Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2504 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,11,16,12,17,10,33)(7,18,23,37,22,36,21,13)(8,24,26,14,28,41,27,40)(9,29,31,42,32,59,30,15)(19,34,46,53,47,38,52,43)(20,48,51,61,50,35,49,44)(25,55,56,54,58,39,57,63)(45,64,65,60,66,68,67,62);(1,2,3,4)(5,7,10,21,12,22,11,23)(6,13,16,18,17,37,33,36)(8,19,26,46,28,47,27,52)(9,20,30,49,32,50,31,51)(14,38,24,53,40,34,41,43)(15,35,29,44,42,48,59,61)(25,45,56,65,58,66,57,67)(39,62,54,68,55,60,63,64);(2,4)(5,8)(6,14,17,40)(7,19,22,47)(9,25)(10,26)(11,27)(12,28)(13,34)(15,39,42,55)(16,24,33,41)(18,43)(20,45,50,66)(21,46,23,52)(29,54,59,63)(30,56)(31,57)(32,58)(35,60)(36,53)(37,38)(44,64)(48,62)(49,65,51,67)(61,68);(5,9,11,31,12,32,10,30)(6,15,33,59,17,42,16,29)(7,20,23,51,22,50,21,49)(8,25,27,57,28,58,26,56)(13,35,36,61,37,48,18,44)(14,39,41,63,40,55,24,54)(19,45,52,67,47,66,46,65)(34,60,53,68,38,62,43,64)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=ef,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274579);