Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1804 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)(10,11)(13,14,18,19)(15,20,21,16)(17,25,27,22)(23,24,26,28)(30,31)(32,39)(33,41)(42,43);(5,6)(7,9)(8,10)(11,12)(14,19)(16,20)(22,25)(24,28)(29,30,34,31)(32,37,39,35)(33,38,41,36)(40,42,44,43);(6,7)(8,12)(13,15)(14,20)(16,19)(17,26)(18,21)(22,24)(23,27)(25,28)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,42,44,43);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,18,20)(14,21,19,15)(17,24,27,28)(22,26,25,23)(29,32,34,39)(30,35,31,37)(33,40,41,44)(36,42,38,43);(5,8)(6,10)(7,11)(9,12)(13,17,18,27)(14,22,19,25)(15,23,21,26)(16,24,20,28)(29,33,34,41)(30,36,31,38)(32,40,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=g,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5189708);