Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1630 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)(5,6,8,10)(7,11,12,9)(13,14)(15,21)(16,24)(17,18)(19,22)(20,25)(23,26)(27,28)(30,31)(32,39)(36,38)(40,44);(5,6)(7,11)(8,10)(9,12)(14,18)(15,22)(16,25)(26,27)(29,30,34,31)(32,37,39,35)(33,38,41,36)(40,42,44,43);(5,6,8,10)(7,9,12,11)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,27)(25,28)(29,31,34,30)(32,35,39,37)(33,38,41,36)(40,42,44,43);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,21)(16,20)(17,18)(19,22)(23,27)(24,25)(26,28)(29,32,34,39)(30,35,31,37)(33,40,41,44)(36,42,38,43);(5,7,8,12)(6,9,10,11)(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)(29,33,34,41)(30,36,31,38)(32,40,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=gh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4376867);