Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2221 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)(13,14,20,17)(15,24,29,26)(16,22,32,30)(18,23,19,34)(21,27,33,28)(25,39,42,40)(31,37,35,38)(36,43,44,41);(6,9)(7,11)(13,15,18,27,20,29,19,28)(14,21,34,26,17,33,23,24)(16,25,35,43,32,42,31,41)(22,36,37,40,30,44,38,39);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,22)(15,25)(17,30)(18,35)(19,31)(20,32)(21,36)(23,38)(24,39)(26,40)(27,43)(28,41)(29,42)(33,44)(34,37);(6,9)(10,12)(13,17,20,14)(15,26,29,24)(16,30,32,22)(18,34,19,23)(21,28,33,27)(25,40,42,39)(31,38,35,37)(36,41,44,43);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,23,17,34)(15,27,29,28)(16,31,32,35)(21,24,33,26)(22,37,30,38)(25,41,42,43)(36,40,44,39)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,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, 7050958);