Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| 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 | |
1614 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)(7,11)(10,12)(13,14)(15,19)(16,18)(17,20)(21,26)(22,23)(24,25)(27,28)(29,30,34,38)(31,36,32,35)(33,41,42,37)(39,43,40,44);(6,9)(10,12)(14,16)(19,23)(20,24)(26,28)(29,31,34,32)(30,35,38,36)(33,39,42,40)(37,43,41,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,22)(14,19,16,23)(17,21,25,27)(20,26,24,28)(29,30)(31,36)(32,35)(33,37)(34,38)(39,44)(40,43)(41,42);(1,2,3,4)(13,16,18,14)(15,19,22,23)(17,24,25,20)(21,26,27,28)(29,32,34,31)(30,36,38,35)(33,40,42,39)(37,44,41,43);(5,7)(6,10)(8,11)(9,12)(13,17,18,25)(14,20,16,24)(15,21,22,27)(19,26,23,28)(29,33,34,42)(30,37,38,41)(31,39,32,40)(35,43,36,44)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4371268);