Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1691 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(8,16)(11,17)(15,20)(18,19)(21,22,28,33)(23,34,38,52)(24,39,41,30)(25,31,35,51)(26,32,27,44)(29,36,48,37)(40,49,42,47)(43,46,45,50);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,37,28,38,26,36)(22,29,32,34,33,48,44,52)(24,35,40,45,41,25,42,43)(30,46,49,31,39,50,47,51);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,30)(23,35)(25,38)(26,42)(27,40)(28,41)(29,46)(31,34)(32,49)(33,39)(36,43)(37,45)(44,47)(48,50)(51,52);(7,14)(10,13)(15,20)(18,19)(21,25,28,35)(22,31,33,51)(23,24,38,41)(26,43,27,45)(29,47,48,49)(30,34,39,52)(32,46,44,50)(36,40,37,42);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,28,27)(22,32,33,44)(23,36,38,37)(24,40,41,42)(25,43,35,45)(29,34,48,52)(30,47,39,49)(31,46,51,50)Polycyclic presentation:
2^9|a^2=g,b^2=fh,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089912);