Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45388 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,28)(24,34)(25,30)(26,40)(27,31)(29,36)(32,44)(33,41)(35,48)(37,43)(38,45)(39,46)(42,49)(47,51)(50,52);(2,3)(7,13)(8,16)(10,17)(11,15)(21,23,27,26)(22,28,31,40)(24,32,37,36)(25,33,39,38)(29,34,44,43)(30,41,46,45)(35,47,50,49)(42,48,51,52);(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,24,23,32,27,37,26,36)(22,29,40,43,31,44,28,34)(25,35,33,47,39,50,38,49)(30,42,45,52,46,51,41,48);(1,3,4,2)(7,13)(10,17)(14,19)(18,20)(21,25)(22,30)(23,33)(24,35)(26,38)(27,39)(28,41)(29,42)(31,46)(32,47)(34,48)(36,49)(37,50)(40,45)(43,52)(44,51);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,23,27,26)(22,28,31,40)(24,36,37,32)(25,38,39,33)(29,43,44,34)(30,45,46,41)(35,47,50,49)(42,48,51,52)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=ghi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104088);