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 | |
1820 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)(8,9)(10,12)(13,14,18,19)(15,17,21,27)(16,23,26,20)(22,28,24,25)(29,30,34,37)(31,35,32,38)(33,36,41,43)(39,42,40,44);(6,8)(7,12)(14,19)(17,27)(20,23)(25,28)(29,31,34,32)(30,35,37,38)(33,39,41,40)(36,42,43,44);(5,7,9,12)(6,10,8,11)(13,15,18,21)(14,17,19,27)(16,24,26,22)(20,28,23,25)(29,32,34,31)(30,35,37,38)(33,40,41,39)(36,42,43,44);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16,18,26)(14,20,19,23)(15,22,21,24)(17,25,27,28)(29,30)(31,38)(32,35)(33,36)(34,37)(39,44)(40,42)(41,43);(5,8,9,6)(7,11,12,10)(13,17)(14,21)(15,19)(16,25)(18,27)(20,24)(22,23)(26,28)(29,33,34,41)(30,36,37,43)(31,39,32,40)(35,42,38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fgh,d^2=gi,e^2=g,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, 4408177);