Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,27,31)(23,33,36,51)(24,37,41,46)(25,30,38,45)(26,44,39,32)(28,40,47,35)(29,42,48,34)(43,50,52,49);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,26,35,27,36,39,40)(22,28,32,33,31,47,44,51)(24,38,34,52,41,25,42,43)(29,30,37,49,48,45,46,50);(7,13)(10,17)(14,19)(18,20)(21,24,39,42,27,41,26,34)(22,29,44,46,31,48,32,37)(23,25,40,52,36,38,35,43)(28,45,51,49,47,30,33,50);(7,13)(10,17)(14,19)(18,20)(21,25)(22,30)(23,34)(24,40)(26,43)(27,38)(28,46)(29,33)(31,45)(32,49)(35,41)(36,42)(37,47)(39,52)(44,50)(48,51);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,31)(24,41)(25,38)(28,47)(32,44)(33,51)(34,42)(43,52)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gi,e^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=dh,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6404934);