Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 | |
2209 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)(13,14,19,25,20,26,18,24)(15,27,29,23,30,39,17,21)(16,31,34,42,35,22,37,41)(28,44,33,38,32,43,36,40);(5,6)(7,11)(8,10)(9,12)(13,15,20,30)(14,21,26,23)(16,32,35,28)(17,18,29,19)(22,40,31,38)(24,27,25,39)(33,37,36,34)(41,43,42,44);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,22)(15,28)(17,36)(18,37)(19,34)(20,35)(21,38)(23,40)(24,42)(25,41)(26,31)(27,43)(29,33)(30,32)(39,44);(7,12)(9,11)(13,17,19,15,20,29,18,30)(14,23,25,39,26,21,24,27)(16,33,34,32,35,36,37,28)(22,38,41,43,31,40,42,44);(6,10)(9,11)(13,18,20,19)(14,24,26,25)(15,17,30,29)(16,34,35,37)(21,39,23,27)(22,41,31,42)(28,33,32,36)(38,43,40,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,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=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6929048);