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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1821 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)(7,11)(10,12)(13,14,20,23,19,24,18,17)(15,25,36,21,29,35,28,27)(16,22,32,40,34,39,33,31)(26,41,42,37,30,44,43,38);(6,9)(7,11)(13,15,19,29)(14,21,24,27)(16,30,34,26)(17,35,23,25)(18,36,20,28)(22,38,39,37)(31,41,40,44)(32,42,33,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,22)(15,26)(17,31)(18,33)(19,34)(20,32)(21,37)(23,40)(24,39)(25,41)(27,38)(28,43)(29,30)(35,44)(36,42);(6,9)(10,12)(13,17,18,24,19,23,20,14)(15,27,28,35,29,21,36,25)(16,31,33,39,34,40,32,22)(26,38,43,44,30,37,42,41);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,23,24,17)(15,28,29,36)(16,32,34,33)(21,35,27,25)(22,31,39,40)(26,42,30,43)(37,41,38,44)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,d^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,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, 7058651);