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 | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1910 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)(23,33)(24,37)(25,30)(26,43)(27,32)(28,35)(29,34)(31,38)(36,50)(39,46)(40,45)(41,44)(42,49)(47,51)(48,52);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,26,35,27,36,38,39)(22,28,31,33,32,46,43,50)(24,25,34,42,40,41,51,52)(29,44,37,49,47,30,45,48);(7,13)(10,17)(14,19)(18,20)(21,24,38,51,27,40,26,34)(22,29,43,45,32,47,31,37)(23,25,39,52,36,41,35,42)(28,44,50,48,46,30,33,49);(7,13)(10,17)(14,19)(18,20)(21,25)(22,30)(23,34)(24,39)(26,42)(27,41)(28,45)(29,33)(31,48)(32,44)(35,40)(36,51)(37,46)(38,52)(43,49)(47,50);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(24,40)(25,41)(29,47)(30,44)(34,51)(37,45)(42,52)(48,49)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gi,e^2=h,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, 6404931);