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 | |
1618 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)(7,11)(10,12)(13,14,18,19)(15,20,21,16)(17,22,26,27)(23,28,25,24)(29,30)(31,36)(32,38)(33,35)(34,37)(39,43)(40,44)(41,42);(6,9)(10,12)(14,19)(16,20)(22,27)(24,28)(29,31,35,32)(30,36,33,38)(34,40,42,39)(37,44,41,43);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,19)(17,25)(18,21)(22,24)(23,26)(27,28)(29,32,35,31)(30,36,33,38)(34,40,42,39)(37,43,41,44);(1,2,3,4)(13,16,18,20)(14,21,19,15)(17,24,26,28)(22,25,27,23)(29,33,35,30)(31,38,32,36)(34,41,42,37)(39,44,40,43);(5,7)(6,10)(8,11)(9,12)(13,17)(14,22)(15,23)(16,24)(18,26)(19,27)(20,28)(21,25)(29,34,35,42)(30,37,33,41)(31,39,32,40)(36,43,38,44)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=i,e^2=g,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, 4987852);