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 | |
1854 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)(6,7)(8,12)(13,14,17,18)(15,19,21,26)(16,23,25,20)(22,28,24,27)(32,39)(35,42)(37,43)(40,44);(6,7)(10,11)(14,18)(15,21)(20,23)(22,24)(29,30,33,36)(31,34,38,41)(32,35,39,42)(37,40,43,44);(5,6,9,7)(8,10,12,11)(13,14)(15,19)(16,20)(17,18)(21,26)(22,27)(23,25)(24,28)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,24)(17,21)(18,26)(20,28)(22,25)(23,27)(29,31,33,38)(30,34,36,41)(32,37,39,43)(35,40,42,44);(5,8,9,12)(6,10,7,11)(13,16)(14,20)(15,22)(17,25)(18,23)(19,27)(21,24)(26,28)(29,32)(30,35)(31,37)(33,39)(34,40)(36,42)(38,43)(41,44)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=fg,d^2=fh,e^2=fg,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, 4775940);