Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1766 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)(23,32)(24,29)(25,30)(26,41)(27,31)(28,35)(33,49)(34,43)(36,48)(37,52)(38,45)(39,46)(40,47)(42,50)(44,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,27,36)(22,28,31,43)(24,37,39,33)(25,35,26,34)(29,44,46,42)(30,48,41,32)(38,50,40,51)(45,52,47,49);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(21,24,27,39)(22,29,31,46)(23,33,36,37)(25,40,26,38)(28,42,43,44)(30,47,41,45)(32,49,48,52)(34,51,35,50);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,31,41)(23,34,36,35)(24,38,39,40)(28,32,43,48)(29,45,46,47)(33,50,37,51)(42,52,44,49);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,31)(28,43)(29,46)(30,41)(32,48)(42,44)(45,47)(49,52)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=i,e^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,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405618);