Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45196 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1716 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)(21,22,28,44,27,32,26,33)(23,34,43,29,36,49,25,31)(24,37,45,48,40,30,41,52)(35,51,39,47,38,50,42,46);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23)(22,29)(24,38)(25,28)(26,43)(27,36)(30,47)(31,32)(33,49)(34,44)(35,40)(37,46)(39,45)(41,42)(48,50)(51,52);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,30)(23,35)(25,42)(26,41)(27,40)(28,45)(29,46)(31,47)(32,37)(33,48)(34,50)(36,38)(39,43)(44,52)(49,51);(7,14)(10,13)(15,20)(18,19)(21,25,28,23,27,43,26,36)(22,31,44,34,32,29,33,49)(24,39,45,38,40,42,41,35)(30,46,52,51,37,47,48,50);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,32)(23,36)(24,40)(25,43)(33,44)(41,45)(46,47)(50,51)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7088460);