Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,10,29,11,16,22,15)(7,17,30,34,21,43,20,12)(8,13,35,38,25,39,24,57)(9,14,31,41,28,42,27,44)(18,47,49,36,50,45,58,32)(19,51,53,33,54,46,55,64)(23,37,26,60,56,66,59,40)(48,67,61,62,68,63,52,65);(1,2)(3,4)(5,7,10,30,11,21,22,20)(6,12,29,17,16,34,15,43)(8,18,35,49,25,50,24,58)(9,19,31,53,28,54,27,55)(13,36,38,45,39,32,57,47)(14,33,41,46,42,64,44,51)(23,48,26,61,56,68,59,52)(37,65,60,67,66,62,40,63);(2,4)(5,8,22,35,11,25,10,24)(6,13,29,57,16,39,15,38)(7,18,30,58,21,50,20,49)(9,26,31,23,28,59,27,56)(12,32,43,47,34,36,17,45)(14,40,44,37,42,60,41,66)(19,52,55,48,54,61,53,68)(33,63,46,62,64,67,51,65);(5,9)(6,14)(7,19)(8,23)(10,31)(11,28)(12,33)(13,37)(15,44)(16,42)(17,46)(18,48)(20,55)(21,54)(22,27)(24,59)(25,56)(26,35)(29,41)(30,53)(32,62)(34,64)(36,65)(38,60)(39,66)(40,57)(43,51)(45,67)(47,63)(49,61)(50,68)(52,58)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364101);