Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1601 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22)(23,31)(24,35)(25,29)(26,40)(27,33)(28,38)(30,39)(32,46)(34,42)(36,50)(37,49)(41,51)(43,47)(44,52)(45,48);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23,25,33)(22,27,29,31)(24,36,37,32)(26,34,39,48)(28,43,44,41)(30,42,40,45)(35,46,49,50)(38,51,52,47);(21,24,26,38,25,37,39,52)(22,28,30,35,29,44,40,49)(23,32,34,47,33,36,48,51)(27,41,42,50,31,43,45,46);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,33)(27,31)(32,36)(34,48)(41,43)(42,45)(46,50)(47,51);(1,3,4,2)(6,12)(10,13)(11,20)(18,19)(22,29)(23,33)(24,37)(30,40)(34,48)(38,52)(41,43)(46,50)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6311495);