Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1600 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)(5,6,9,12)(7,8,11,10)(13,14)(15,18)(16,21)(17,20)(19,23)(22,25)(24,27)(26,28)(33,41)(36,40)(38,44)(42,43);(5,7,9,11)(6,10,12,8)(14,15)(16,23)(20,22)(24,28)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(7,11)(8,10)(13,15,18,14)(16,21,23,19)(17,20,25,22)(24,26,28,27)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,43,44,42);(5,8,9,10)(6,11,12,7)(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)(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=gh,c^2=fgh,d^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4575761);