Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| 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 | #37553 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1757 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(5,6)(7,10)(8,15)(9,11)(12,16)(13,17)(14,20)(18,19)(21,22)(23,28)(24,36)(25,30)(26,40)(27,31)(29,37)(32,46)(33,42)(34,43)(35,50)(38,44)(39,45)(41,47)(48,51)(49,52);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23)(22,28)(24,32)(25,33)(26,35)(27,34)(29,41)(30,42)(31,43)(36,46)(37,47)(38,48)(39,49)(40,50)(44,51)(45,52);(6,11)(7,13)(12,15)(14,19)(21,24,26,37,27,38,25,39)(22,29,30,36,31,45,40,44)(23,32,35,47,34,48,33,49)(28,41,42,46,43,52,50,51);(7,13)(10,17)(14,19)(18,20)(21,25,27,26)(22,30,31,40)(23,33,34,35)(24,37,38,39)(28,42,43,50)(29,44,45,36)(32,47,48,49)(41,51,52,46);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(23,34)(24,38)(28,43)(29,45)(33,35)(36,44)(37,39)(42,50)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405758);