Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469971 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1683 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,3)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,26,29)(23,31,33,46)(24,28,36,45)(25,38,40,30)(27,39,42,34)(32,47,35,49)(37,52,51,44)(41,50,43,48);(1,2,4,3)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23)(22,27)(24,35)(25,39)(26,33)(28,43)(29,42)(30,46)(31,38)(32,36)(34,40)(37,48)(41,45)(44,47)(49,52)(50,51);(1,3,4,2)(7,13)(10,17)(14,15)(18,19)(21,24)(22,28)(23,32)(25,37)(26,36)(27,41)(29,45)(30,44)(31,47)(33,35)(34,48)(38,52)(39,50)(40,51)(42,43)(46,49);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(23,33)(27,42)(31,46)(32,35)(34,39)(41,43)(47,49)(48,50);(6,12)(10,17)(11,20)(18,19)(22,29)(24,36)(27,42)(30,38)(31,46)(32,35)(37,51)(48,50)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6391949);