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)(23,33)(24,28)(25,30)(26,41)(27,32)(29,36)(31,34)(35,47)(37,42)(38,43)(39,45)(40,51)(44,49)(46,48)(50,52);(1,2,3,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,34,38,27,36,26,24)(22,28,41,29,32,43,31,33)(25,35,48,50,39,49,40,37)(30,42,51,44,45,52,46,47);(7,13)(10,17)(14,19)(18,20)(21,24,26,36,27,38,34,23)(22,29,31,28,32,33,41,43)(25,37,40,49,39,50,48,35)(30,44,46,42,45,47,51,52);(7,13)(10,17)(14,19)(18,20)(21,25)(22,30)(23,35)(24,37)(26,40)(27,39)(28,42)(29,44)(31,46)(32,45)(33,47)(34,48)(36,49)(38,50)(41,51)(43,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(25,39)(30,45)(35,49)(37,50)(40,48)(42,52)(44,47)(46,51)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=g,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, 6404923);