Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2778 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)(21,22,27,32)(23,33,35,44)(24,38,41,45)(25,31,26,30)(28,36,43,37)(29,40,46,39)(34,48,49,42)(47,50,52,51);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,27,35)(22,28,32,43)(24,34,41,49)(25,36,26,37)(29,42,46,48)(30,44,31,33)(38,52,45,47)(39,50,40,51);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24)(22,29)(23,34)(25,39)(26,40)(27,41)(28,42)(30,38)(31,45)(32,46)(33,47)(35,49)(36,50)(37,51)(43,48)(44,52);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,35)(28,43)(33,44)(34,49)(36,37)(42,48)(47,52)(50,51);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,30,32,31)(23,36,35,37)(24,39,41,40)(28,44,43,33)(29,38,46,45)(34,50,49,51)(42,52,48,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436338);