Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,29,42)(16,21,32,40)(17,23,18,22)(20,34,37,28)(26,43,30,36)(27,38,44,35)(31,41,33,39);(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,20,24,37)(16,30,32,26)(17,28,18,34)(21,38,40,35)(22,25,23,42)(27,31,44,33)(36,39,43,41);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,32)(14,21,24,40)(15,26,29,30)(17,33,18,31)(20,35,37,38)(22,41,23,39)(25,43,42,36)(27,28,44,34);(7,12)(9,11)(13,16)(14,21)(15,27)(17,31)(18,33)(19,32)(20,36)(22,39)(23,41)(24,40)(25,38)(26,34)(28,30)(29,44)(35,42)(37,43);(6,10)(9,11)(13,17,19,18)(14,22,24,23)(15,28,29,34)(16,31,32,33)(20,25,37,42)(21,39,40,41)(26,27,30,44)(35,36,38,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,f^b=fi,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6901492);