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 | |
1598 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)(9,12)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30)(31,36)(32,33)(34,35)(37,41)(38,39)(40,44)(42,43);(5,6,8,10)(7,9,11,12)(15,22)(16,25)(18,21)(19,24)(30,35)(32,41)(36,44)(38,42);(5,7,8,11)(6,9,10,12)(13,15)(14,18)(16,23)(17,22)(19,26)(20,21)(24,27)(25,28)(29,31,34,40)(30,36,35,44)(32,42,41,38)(33,43,37,39);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,14)(15,21)(16,19)(17,20)(18,22)(23,27)(24,25)(26,28)(29,32)(30,37)(31,38)(33,35)(34,41)(36,43)(39,44)(40,42);(5,6)(7,9)(8,10)(11,12)(13,16)(14,19)(15,23)(17,25)(18,26)(20,24)(21,27)(22,28)(29,33,34,37)(30,32,35,41)(31,39,40,43)(36,38,44,42)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=fgh,d^2=fgi,e^2=f,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, 4989862);