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 | #45245 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2027 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)(13,14,19,34,18,23,17,24)(15,25,33,20,27,21,28,37)(16,22,35,44,31,41,32,40)(26,39,43,36,30,38,29,42);(5,6)(7,11)(8,10)(9,12)(13,15,18,27)(14,20,23,37)(16,29,31,43)(17,33,19,28)(21,34,25,24)(22,39,41,38)(26,35,30,32)(36,44,42,40);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,30)(17,28)(18,27)(19,33)(20,24)(22,39)(23,25)(26,31)(29,35)(32,43)(34,37)(36,44)(38,41)(40,42);(7,12)(9,11)(13,16,19,35,18,31,17,32)(14,22,34,44,23,41,24,40)(15,26,33,43,27,30,28,29)(20,36,21,38,37,42,25,39);(6,10)(9,11)(14,23)(15,27)(16,31)(24,34)(28,33)(32,35)(36,42)(38,39)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=fi,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=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, 6913981);