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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(7,11)(10,12)(13,14)(15,20)(16,26)(17,22)(18,32)(19,23)(21,29)(24,35)(25,33)(27,38)(28,39)(30,36)(31,42)(34,40)(37,43)(41,44);(2,3)(6,9)(7,11)(13,15,19,18)(14,20,23,32)(16,24,28,29)(17,25,30,31)(21,26,35,39)(22,33,36,42)(27,37,41,40)(34,38,43,44);(5,6)(7,10)(8,9)(11,12)(13,16,18,29,19,28,15,24)(14,21,20,26,23,35,32,39)(17,27,31,40,30,41,25,37)(22,34,33,38,36,43,42,44);(1,3,4,2)(6,9)(10,12)(13,17)(14,22)(15,25)(16,27)(18,31)(19,30)(20,33)(21,34)(23,36)(24,37)(26,38)(28,41)(29,40)(32,42)(35,43)(39,44);(5,7,8,11)(6,10,9,12)(16,28)(17,30)(21,35)(22,36)(24,29)(25,31)(26,39)(33,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=f,e^2=h,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, 7063450);