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 | |
1744 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)(5,6,9,10)(7,12,8,11)(13,14)(15,20)(16,24)(17,22)(18,19)(21,26)(23,27)(25,28)(30,31)(32,39)(36,38)(40,44);(6,10)(7,8)(14,19)(16,26)(20,28)(22,23)(29,30,34,31)(32,37,39,35)(33,38,41,36)(40,42,44,43);(5,7,9,8)(6,11,10,12)(13,15,18,25)(14,20,19,28)(16,22,26,23)(17,21,27,24)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,42,44,43);(1,2,3,4)(5,8,9,7)(6,11,10,12)(13,16)(14,21)(15,23)(17,20)(18,26)(19,24)(22,25)(27,28)(29,32)(30,35)(31,37)(33,40)(34,39)(36,42)(38,43)(41,44);(5,6)(7,12)(8,11)(9,10)(13,17)(14,22)(15,24)(16,20)(18,27)(19,23)(21,25)(26,28)(29,33,34,41)(30,36,31,38)(32,40,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=f,d^2=fi,e^2=gh,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, 4991085);