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) = 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 | |
1708 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)(5,6,8,11)(7,9,12,10)(13,14)(15,20)(16,17)(18,19)(21,25)(22,28)(23,24)(26,27)(30,31)(32,39)(33,41)(42,43);(5,6)(7,9)(8,11)(10,12)(14,19)(15,22)(16,25)(24,27)(29,30,34,31)(32,37,39,35)(33,38,41,36)(40,42,44,43);(5,7,8,12)(6,9,11,10)(13,15,18,22)(14,20,19,28)(16,26,25,23)(17,27,21,24)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,42,44,43);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,16)(14,21)(15,23)(17,19)(18,25)(20,27)(22,26)(24,28)(29,32,34,39)(30,35,31,37)(33,40,41,44)(36,42,38,43);(5,6)(7,10)(8,11)(9,12)(13,17,18,21)(14,16,19,25)(15,24,22,27)(20,23,28,26)(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=f,d^2=fgi,e^2=gh,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, 5190344);