Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499792 |
| 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×#3565 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2760 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)(21,22)(23,32)(24,37)(25,30)(26,31)(27,40)(28,34)(29,38)(33,48)(35,47)(36,43)(39,44)(41,45)(42,51)(46,50)(49,52);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23)(22,28)(24,33)(25,35)(26,34)(27,36)(29,42)(30,43)(31,32)(37,52)(38,50)(39,49)(40,47)(41,51)(44,48)(45,46);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24)(22,29)(23,33)(25,38)(26,39)(27,41)(28,42)(30,44)(31,45)(32,46)(34,49)(35,50)(36,51)(37,40)(43,48)(47,52);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,34)(28,32)(33,49)(35,36)(42,46)(43,47)(48,52)(50,51);(6,12)(10,13)(11,20)(18,19)(21,25,26,27)(22,30,31,40)(23,35,34,36)(24,38,39,41)(28,43,32,47)(29,44,45,37)(33,50,49,51)(42,48,46,52)Polycyclic presentation:
2^9|a^2=fi,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436433);