Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1852 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)(5,6,8,10)(7,9,11,12)(13,14,18,19)(15,16,20,24)(17,25,26,21)(22,28,27,23)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(7,11)(9,12)(14,19)(15,20)(17,26)(23,28)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(5,7)(6,9)(8,11)(10,12)(13,15)(14,16)(17,22)(18,20)(19,24)(21,23)(25,28)(26,27)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44);(1,2)(3,4)(13,16)(14,20)(15,19)(17,23)(18,24)(21,27)(22,25)(26,28)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,6)(7,9)(8,10)(11,12)(13,17,18,26)(14,21,19,25)(15,22,20,27)(16,23,24,28)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4373332);