Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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 | |
1674 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)(5,6,8,10)(7,9,11,12)(13,14)(15,17)(16,24)(18,19)(20,23)(21,27)(22,28)(25,26)(33,41)(36,40)(38,44)(42,43);(14,19)(17,20)(21,24)(26,28)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,25)(17,19)(18,23)(21,26)(22,27)(24,28)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,27)(14,21,19,24)(15,22,23,25)(17,26,20,28)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,43,44,42);(5,7,8,11)(6,9,10,12)(13,17,18,20)(14,15,19,23)(16,26,27,28)(21,22,24,25)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=h,d^2=fghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4250115);