Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1612 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)(5,6,9,11)(7,8,12,10)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30,34,35)(31,36,41,44)(32,33,38,37)(39,40,42,43);(5,7,9,12)(6,10,11,8)(15,22)(16,25)(18,21)(19,24)(30,35)(32,38)(36,44)(39,42);(5,6,9,11)(7,8,12,10)(13,15,17,22)(14,18,20,21)(16,23,25,28)(19,26,24,27)(29,31,34,41)(30,36,35,44)(32,42,38,39)(33,43,37,40);(1,2,3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,21)(16,19)(17,20)(18,22)(23,27)(24,25)(26,28)(29,32,34,38)(30,37,35,33)(31,39,41,42)(36,43,44,40);(5,8)(6,7)(9,10)(11,12)(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,33,34,37)(30,38,35,32)(31,40,41,43)(36,42,44,39)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fg,d^2=hi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4991936);