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 | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
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)(26,32)(28,40)(30,34)(37,41)(39,44)(42,43);(5,6)(7,11)(8,10)(9,12)(13,14,19,23)(15,24,27,21)(16,28,31,36)(17,20,18,33)(22,34,40,30)(25,41,29,44)(26,35,32,38)(37,43,39,42);(2,4)(5,7,8,12)(6,9,10,11)(13,15,19,27)(14,21,23,24)(16,29,31,25)(17,32,18,26)(20,35,33,38)(22,39,40,37)(28,41,36,44)(30,42,34,43);(7,12)(9,11)(13,16,19,31)(14,22,23,40)(15,25,27,29)(17,30,18,34)(20,36,33,28)(21,37,24,39)(26,42,32,43)(35,41,38,44);(6,10)(9,11)(13,17,19,18)(14,20,23,33)(15,26,27,32)(16,30,31,34)(21,38,24,35)(22,36,40,28)(25,42,29,43)(37,44,39,41)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,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=ehi,f^b=fi,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6969878);