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 | #2431 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2012 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)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22,27,32)(23,33,37,47)(24,38,39,52)(25,30,41,45)(26,43,34,31)(28,44,46,36)(29,35,48,40)(42,50,51,49);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,34,44,27,37,26,36)(22,28,43,47,32,46,31,33)(24,25,35,51,39,41,40,42)(29,45,52,49,48,30,38,50);(6,11)(7,13)(12,15)(14,19)(21,24,26,40,27,39,34,35)(22,29,31,38,32,48,43,52)(23,25,36,42,37,41,44,51)(28,45,33,50,46,30,47,49);(7,13)(10,17)(14,19)(18,20)(21,25)(22,30)(23,35)(24,36)(26,42)(27,41)(28,38)(29,47)(31,49)(32,45)(33,48)(34,51)(37,40)(39,44)(43,50)(46,52);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(24,39)(25,41)(29,48)(30,45)(35,40)(38,52)(42,51)(49,50)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gh,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, 6406124);