Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3455 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,27,32)(23,33,35,48)(24,29,38,45)(25,31,26,30)(28,41,43,36)(34,49,37,51)(39,47,40,46)(42,52,44,50);(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23)(22,28)(24,37)(25,36)(26,41)(27,35)(29,44)(30,33)(31,48)(32,43)(34,38)(39,52)(40,50)(42,45)(46,51)(47,49);(2,4)(7,13)(10,17)(14,15)(18,19)(21,24)(22,29)(23,34)(25,39)(26,40)(27,38)(28,42)(30,46)(31,47)(32,45)(33,49)(35,37)(36,50)(41,52)(43,44)(48,51);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(23,35)(24,38)(28,43)(29,45)(33,48)(36,41)(39,40)(46,47);(6,12)(10,17)(11,20)(18,19)(21,25,27,26)(22,30,32,31)(23,36,35,41)(24,39,38,40)(28,33,43,48)(29,46,45,47)(34,50,37,52)(42,49,44,51)Polycyclic presentation:
2^9|a^2=i,d^2=hi,e^2=i,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, 6388719);