Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6131635 |
| automorphism | Aut(G) order 4194304 |
| 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 | |
1908 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)(7,11)(10,12)(15,21)(16,23)(18,24)(19,26)(30,34)(32,40)(35,37)(38,44);(1,2)(3,4)(6,9)(7,11)(15,21)(18,24)(22,27)(25,28)(29,30,33,34)(31,37,39,35)(32,41,40,36)(38,42,44,43);(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,18,21,24)(16,19,23,26)(22,25,27,28)(29,30)(31,35)(32,36)(33,34)(37,39)(38,42)(40,41)(43,44);(5,7,8,11)(6,10,9,12)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28)(29,31,33,39)(30,35,34,37)(32,38,40,44)(36,42,41,43);(13,16)(14,19)(15,22)(17,23)(18,25)(20,26)(21,27)(24,28)(29,32,33,40)(30,36,34,41)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|b^2=g,c^2=ghi,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6039777);