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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1774 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)(23,31)(24,28)(25,37)(26,29)(27,33)(30,39)(32,45)(34,44)(35,42)(36,49)(38,41)(40,47)(43,48)(46,52)(50,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,18)(15,20)(16,19)(21,23,26,34)(22,27,29,41)(24,32,35,46)(25,38,39,33)(28,40,42,51)(30,44,37,31)(36,50,48,47)(43,52,49,45);(1,2,3,4)(6,11)(7,13)(12,15)(14,19)(21,24)(22,28)(23,32)(25,36)(26,35)(27,40)(29,42)(30,43)(31,45)(33,47)(34,46)(37,49)(38,50)(39,48)(41,51)(44,52);(7,13)(10,17)(14,19)(18,20)(22,29)(27,41)(28,42)(30,37)(31,44)(40,51)(43,49)(45,52);(5,8,9,16)(6,12,11,15)(7,14,13,19)(10,18,17,20)(24,35)(28,42)(32,46)(36,48)(40,51)(43,49)(45,52)(47,50)Polycyclic presentation:
2^9|b^2=i,c^2=gh,e^2=h,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,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405638);