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 | |
1893 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)(13,14,17,20)(15,21,22,18)(16,19,24,26)(23,27,28,25)(29,30)(31,36)(32,40)(33,42)(34,35)(37,43)(38,39)(41,44);(5,6,8,10)(7,9,12,11)(15,22)(18,21)(23,28)(25,27)(30,35)(31,37)(33,40)(38,41);(7,12)(9,11)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)(29,31,34,37)(30,36,35,43)(32,41,42,38)(33,39,40,44);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,23)(17,22)(19,25)(20,21)(24,28)(26,27)(29,32)(30,33)(31,38)(34,42)(35,40)(36,44)(37,41)(39,43);(5,6)(7,9)(8,10)(11,12)(13,16,17,24)(14,19,20,26)(15,23,22,28)(18,25,21,27)(29,33)(30,32)(31,39)(34,40)(35,42)(36,41)(37,44)(38,43)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=h,d^2=fgi,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, 4408102);