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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,7)(8,12)(13,14)(15,16)(17,21)(18,19)(20,24)(22,23)(25,28)(26,27)(29,30,34,35)(31,38,41,36)(32,42,37,33)(39,43,44,40);(5,6,9,7)(8,10,12,11)(14,19)(15,20)(21,27)(22,25)(30,35)(32,37)(36,38)(39,44);(8,12)(10,11)(13,15,18,20)(14,16,19,24)(17,25,26,22)(21,28,27,23)(29,31,34,41)(30,36,35,38)(32,39,37,44)(33,43,42,40);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16)(14,20)(15,19)(17,23)(18,24)(21,25)(22,27)(26,28)(29,32,34,37)(30,33,35,42)(31,39,41,44)(36,43,38,40);(5,8,9,12)(6,10,7,11)(13,17,18,26)(14,21,19,27)(15,22,20,25)(16,23,24,28)(29,33)(30,37)(31,40)(32,35)(34,42)(36,39)(38,44)(41,43)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=gh,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, 4575758);