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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2004 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)(7,11)(8,10)(9,12)(15,21)(16,23)(19,24)(20,25)(29,30,34,37)(31,32,38,35)(33,41,42,36)(39,43,44,40);(5,6,8,10)(7,9,12,11)(14,18)(19,24)(20,25)(26,28)(31,38)(32,35)(39,44)(40,43);(7,12)(9,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,35)(32,37)(33,39)(34,38)(36,43)(40,41)(42,44);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32,34,35)(30,31,37,38)(33,40,42,43)(36,39,41,44);(5,6)(7,9)(8,10)(11,12)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=gh,d^2=h,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, 4376326);