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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
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)(23,33)(24,29)(25,32)(26,30)(27,31)(28,36)(34,49)(35,44)(37,43)(38,52)(39,48)(40,47)(41,46)(42,50)(45,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,27,37)(22,28,31,44)(24,38,40,34)(25,36,26,35)(29,45,47,42)(30,33,32,43)(39,50,41,51)(46,49,48,52);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(21,24,27,40)(22,29,31,47)(23,34,37,38)(25,41,26,39)(28,42,44,45)(30,48,32,46)(33,49,43,52)(35,51,36,50);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,31,32)(23,35,37,36)(24,39,40,41)(28,43,44,33)(29,46,47,48)(34,50,38,51)(42,49,45,52);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(22,31)(28,44)(29,47)(30,32)(33,43)(42,45)(46,48)(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,d^a=di,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, 6405627);