Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1626 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)(7,10)(8,11)(9,12)(13,14,18,16)(15,21,22,19)(17,24,25,20)(23,26,28,27)(30,32)(33,41)(35,39)(37,43);(5,7)(6,8)(9,11)(10,12)(15,22)(17,25)(19,21)(20,24)(29,30,34,32)(31,35,38,39)(33,36,41,40)(37,42,43,44);(5,7,9,11)(6,10,12,8)(13,15,18,22)(14,19,16,21)(17,23,25,28)(20,26,24,27)(29,31)(30,35)(32,39)(33,37)(34,38)(36,42)(40,44)(41,43);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,16,18,14)(15,19,22,21)(17,24,25,20)(23,26,28,27)(29,32,34,30)(31,35,38,39)(33,36,41,40)(37,44,43,42);(5,8,9,10)(6,11,12,7)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)(29,33)(30,36)(31,37)(32,40)(34,41)(35,42)(38,43)(39,44)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=h,d^2=fh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5544269);