Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470901 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1630 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)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,28,34)(23,35,40,47)(24,41,42,50)(25,31,36,46)(26,33,27,32)(29,39,48,38)(30,43,49,37)(44,52,45,51);(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23,27,39,28,40,26,38)(22,29,33,47,34,48,32,35)(24,36,37,44,42,25,43,45)(30,46,41,51,49,31,50,52);(2,4)(7,13)(10,17)(14,15)(18,19)(21,24,26,43,28,42,27,37)(22,30,32,50,34,49,33,41)(23,36,38,45,40,25,39,44)(29,46,35,52,48,31,47,51);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(21,25,28,36)(22,31,34,46)(23,37,40,43)(24,39,42,38)(26,44,27,45)(29,41,48,50)(30,47,49,35)(32,51,33,52);(6,12)(10,17)(11,20)(18,19)(24,42)(25,36)(30,49)(31,46)(37,43)(41,50)(44,45)(51,52)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,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, 6389453);