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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1885 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)(23,29)(24,36)(25,30)(26,41)(27,32)(28,34)(31,37)(33,43)(35,44)(38,49)(39,45)(40,51)(42,48)(46,50)(47,52);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,26,34,27,35,37,24)(22,28,31,29,32,36,41,44)(25,33,40,48,39,47,50,38)(30,42,46,43,45,49,51,52);(7,13)(10,17)(14,19)(18,20)(21,24,37,35,27,34,26,23)(22,29,41,28,32,44,31,36)(25,38,50,47,39,48,40,33)(30,43,51,42,45,52,46,49);(7,13)(10,17)(14,19)(18,20)(21,25)(22,30)(23,33)(24,38)(26,40)(27,39)(28,42)(29,43)(31,46)(32,45)(34,48)(35,47)(36,49)(37,50)(41,51)(44,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(25,39)(30,45)(33,47)(38,48)(40,50)(42,49)(43,52)(46,51)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^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6404925);