Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1944 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)(6,9)(10,11)(14,18)(15,21)(16,23)(26,28)(30,31)(35,37)(36,38)(43,44);(5,6)(7,10)(8,9)(11,12)(14,18)(15,21)(20,25)(22,27)(29,30,34,31)(32,37,40,35)(33,36,42,38)(39,44,41,43);(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,34,30)(32,37,40,35)(33,38,42,36)(39,44,41,43);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32)(30,35)(31,37)(33,41)(34,40)(36,44)(38,43)(39,42);(5,7)(6,10)(8,12)(9,11)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,42)(30,36,31,38)(32,39,40,41)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fgi,b^2=gh,c^2=gh,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4776079);