Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| 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 | #35737 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2153 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)(7,10)(8,9)(11,12)(13,14,18,23)(15,25,28,20)(16,30,32,21)(17,33,36,22)(19,24,37,35)(26,34,41,43)(27,31,40,44)(29,42,38,39);(6,9)(10,12)(13,15,18,28)(14,20,23,25)(16,31,32,44)(17,34,36,43)(19,38,37,29)(21,40,30,27)(22,41,33,26)(24,42,35,39);(2,4)(13,16)(14,21)(15,26)(17,35)(18,32)(19,22)(20,34)(23,30)(24,36)(25,43)(27,39)(28,41)(29,44)(31,38)(33,37)(40,42);(5,7,8,11)(6,10,9,12)(16,22)(17,30)(19,37)(21,36)(24,35)(26,44)(27,34)(29,38)(31,41)(32,33)(39,42)(40,43);(13,17,18,36)(14,22,23,33)(15,27,28,40)(16,24,32,35)(19,21,37,30)(20,31,25,44)(26,42,41,39)(29,43,38,34)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482431);