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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,24,20,25,18,33)(15,26,28,23,29,36,17,21)(16,22,31,39,32,40,34,38)(27,41,42,37,43,44,30,35);(5,6)(7,11)(8,10)(9,12)(13,15,20,29)(14,21,25,23)(16,27,32,43)(17,18,28,19)(22,35,40,37)(24,36,33,26)(30,34,42,31)(38,41,39,44);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,27)(17,30)(18,34)(19,31)(20,32)(21,35)(23,37)(24,39)(25,40)(26,41)(28,42)(29,43)(33,38)(36,44);(7,12)(9,11)(13,17,19,15,20,28,18,29)(14,23,24,36,25,21,33,26)(16,30,31,27,32,42,34,43)(22,37,39,44,40,35,38,41);(6,10)(9,11)(13,18,20,19)(14,24,25,33)(15,28,29,17)(16,31,32,34)(21,36,23,26)(22,38,40,39)(27,30,43,42)(35,41,37,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,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, 6929076);