Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,18)(16,19)(17,20)(21,22,27,32)(23,33,37,48)(24,29,39,46)(25,30,40,47)(26,41,38,31)(28,42,45,36)(34,49,50,44)(35,43,51,52);(5,7,10,17)(6,11,14,20)(8,16,9,15)(12,19,13,18)(21,23)(22,28)(24,34)(25,35)(26,42)(27,37)(29,43)(30,44)(31,48)(32,45)(33,41)(36,38)(39,50)(40,51)(46,52)(47,49);(2,4)(5,8,10,9)(6,12,14,13)(7,15,17,16)(11,18,20,19)(21,24)(22,29)(23,34)(25,38)(26,40)(27,39)(28,43)(30,31)(32,46)(33,49)(35,36)(37,50)(41,47)(42,51)(44,48)(45,52);(5,9,10,8)(6,13,14,12)(7,16,17,15)(11,19,20,18)(21,25,27,40)(22,30,32,47)(23,35,37,51)(24,38,39,26)(28,44,45,49)(29,31,46,41)(33,43,48,52)(34,36,50,42);(6,14)(11,20)(12,13)(18,19)(24,39)(25,40)(29,46)(30,47)(34,50)(35,51)(43,52)(44,49)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396785);