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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,11)(7,10,12,9)(13,14)(15,21)(16,23)(17,20)(18,19)(22,27)(24,28)(25,26)(30,34)(31,37)(32,40)(42,44);(5,7)(6,9)(8,12)(10,11)(14,19)(16,21)(17,25)(22,28)(29,30,33,34)(31,38,37,35)(32,41,40,36)(39,42,43,44);(5,7,8,12)(6,10,11,9)(13,15,18,23)(14,16,19,21)(17,22,25,28)(20,24,26,27)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,41)(42,43);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,16)(14,15)(17,24)(18,21)(19,23)(20,22)(25,27)(26,28)(29,31)(30,35)(32,39)(33,37)(34,38)(36,42)(40,43)(41,44);(5,6)(7,10)(8,11)(9,12)(13,17,18,25)(14,20,19,26)(15,22,23,28)(16,24,21,27)(29,32,33,40)(30,36,34,41)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=gh,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, 4408175);