Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1948 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)(6,9)(7,11)(13,14)(15,16)(17,21)(18,19)(20,23)(22,24)(25,26)(27,28)(30,34)(32,40)(35,37)(38,44);(7,11)(10,12)(14,19)(16,23)(21,26)(24,28)(29,30,33,34)(31,37,39,35)(32,36,40,41)(38,43,44,42);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,22)(18,20)(19,23)(21,24)(25,27)(26,28)(29,31,33,39)(30,35,34,37)(32,38,40,44)(36,42,41,43);(1,2,3,4)(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,19)(17,24)(18,23)(21,27)(22,26)(25,28)(29,30)(31,37)(32,36)(33,34)(35,39)(38,43)(40,41)(42,44);(13,17,18,25)(14,21,19,26)(15,22,20,27)(16,24,23,28)(29,32,33,40)(30,36,34,41)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4371295);