Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #42796 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1885 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(14,20)(17,18)(21,35)(22,36)(23,33)(24,38)(26,34)(28,40)(30,31)(37,41)(39,44)(42,43);(5,6)(7,11)(8,10)(9,12)(13,14,18,20,19,23,17,33)(15,24,26,38,27,21,34,35)(16,28,31,40,32,36,30,22)(25,41,42,37,29,44,43,39);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,29)(17,34)(18,26)(19,27)(20,35)(22,39)(23,24)(25,32)(28,41)(30,42)(31,43)(33,38)(36,44)(37,40);(7,12)(9,11)(13,16,19,32)(14,22,23,40)(15,25,27,29)(17,31,18,30)(20,36,33,28)(21,37,24,39)(26,43,34,42)(35,41,38,44);(6,10)(9,11)(13,17,19,18)(14,20,23,33)(15,26,27,34)(16,30,32,31)(21,38,24,35)(22,28,40,36)(25,42,29,43)(37,41,39,44)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910390);