Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1692 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)(21,22,27,30)(23,28,34,44)(24,37,39,46)(25,32,26,31)(29,40,45,38)(33,47,48,52)(35,43,36,42)(41,50,51,49);(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23)(22,28)(24,33)(25,35)(26,36)(27,34)(29,41)(30,44)(31,42)(32,43)(37,47)(38,49)(39,48)(40,50)(45,51)(46,52);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24)(22,29)(23,33)(25,38)(26,40)(27,39)(28,41)(30,45)(31,37)(32,46)(34,48)(35,49)(36,50)(42,47)(43,52)(44,51);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,30)(23,34)(29,45)(31,32)(33,48)(35,36)(37,46)(49,50);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,31,30,32)(23,35,34,36)(24,38,39,40)(28,42,44,43)(29,37,45,46)(33,49,48,50)(41,47,51,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436985);