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 | #45332 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2148 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)(13,14,19,23,20,24,18,17)(15,25,28,36,29,40,27,21)(16,22,32,38,33,39,34,31)(26,41,42,37,30,44,43,35);(6,9)(7,11)(13,15,20,29)(14,21,24,36)(16,30,33,26)(17,25,23,40)(18,28,19,27)(22,37,39,35)(31,44,38,41)(32,42,34,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,20,33)(14,22,24,39)(15,26,29,30)(17,31,23,38)(18,34,19,32)(21,35,36,37)(25,41,40,44)(27,43,28,42);(6,9)(10,12)(13,17,18,24,20,23,19,14)(15,21,27,40,29,36,28,25)(16,31,34,39,33,38,32,22)(26,35,43,44,30,37,42,41);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,24,17)(15,27,29,28)(16,32,33,34)(21,25,36,40)(22,31,39,38)(26,42,30,43)(35,44,37,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,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, 7053337);