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 | |
1592 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)(5,6,10,9)(7,12,8,11)(13,14,18,16)(15,21,23,19)(17,20,25,24)(22,27,28,26)(29,30)(31,37)(32,41)(33,42)(34,35)(36,40)(38,39)(43,44);(5,7,10,8)(6,11,9,12)(17,25)(20,24)(22,28)(26,27)(30,35)(33,41)(36,37)(39,43);(7,8)(11,12)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27)(29,31,34,40)(30,36,35,37)(32,38,42,44)(33,39,41,43);(1,2)(3,4)(5,8,10,7)(6,11,9,12)(13,16,18,14)(15,19,23,21)(17,24,25,20)(22,26,28,27)(29,32,34,42)(30,33,35,41)(31,38,40,44)(36,39,37,43);(5,9,10,6)(7,12,8,11)(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)(29,33)(30,32)(31,39)(34,41)(35,42)(36,38)(37,44)(40,43)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=g,d^2=gh,e^2=fg,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, 5524116);