Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| 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 | #28354 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2027 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)(21,22,27,31)(23,29,36,45)(24,37,35,28)(25,30,40,48)(26,42,33,32)(34,44,39,51)(38,46,50,43)(41,52,49,47);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,33,24,27,36,26,35)(22,28,42,45,31,37,32,29)(25,39,49,50,40,34,41,38)(30,46,52,44,48,43,47,51);(6,11)(7,13)(12,15)(14,19)(21,24,33,36,27,35,26,23)(22,29,42,28,31,45,32,37)(25,38,49,39,40,50,41,34)(30,44,52,43,48,51,47,46);(7,13)(10,17)(14,19)(18,20)(21,25)(22,30)(23,34)(24,38)(26,41)(27,40)(28,43)(29,44)(31,48)(32,47)(33,49)(35,50)(36,39)(37,46)(42,52)(45,51);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,31)(25,40)(28,37)(29,45)(32,42)(34,39)(38,50)(41,49)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gh,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^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406116);