Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393982 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62859 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,12,18)(7,19,23,40)(8,24,27,42)(9,15,29,39)(10,17,11,16)(13,30,35,22)(14,32,37,26)(20,33,48,59)(21,49,52,44)(25,54,57,61)(28,41,31,38)(34,51,60,50)(36,55,62,56)(43,46,53,47)(45,64,66,58)(63,65,68,67);(1,2)(3,4)(5,7)(6,13)(8,20)(9,21)(10,30)(11,22)(12,23)(14,33)(15,34)(16,40)(17,19)(18,35)(24,53)(25,45)(26,47)(27,48)(28,51)(29,52)(31,50)(32,46)(36,58)(37,59)(38,44)(39,60)(41,49)(42,43)(54,68)(55,67)(56,65)(57,66)(61,63)(62,64);(2,4)(5,8,12,27)(6,14,18,37)(7,20,23,48)(9,25,29,57)(10,26,11,32)(13,33,35,59)(15,36,39,62)(16,24,17,42)(19,43,40,53)(21,45,52,66)(22,46,30,47)(28,55,31,56)(34,58,60,64)(38,61,41,54)(44,63,49,68)(50,65,51,67);(5,9)(6,15)(7,21)(8,25)(10,31)(11,28)(12,29)(13,34)(14,36)(16,41)(17,38)(18,39)(19,44)(20,45)(22,51)(23,52)(24,54)(26,56)(27,57)(30,50)(32,55)(33,58)(35,60)(37,62)(40,49)(42,61)(43,63)(46,67)(47,65)(48,66)(53,68)(59,64)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,h^b=hi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301181);