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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1667 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,9)(10,11)(13,14,17,20)(15,21,22,18)(16,19,24,26)(23,27,28,25)(29,30,34,35)(31,36,37,32)(33,41,42,38)(39,40,44,43);(5,6,8,9)(7,11,12,10)(15,22)(18,21)(23,28)(25,27)(30,35)(31,37)(33,42)(40,43);(5,6,8,9)(7,10,12,11)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)(29,31)(30,36)(32,35)(33,39)(34,37)(38,43)(40,41)(42,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,23)(17,22)(19,25)(20,21)(24,28)(26,27)(29,32,34,36)(30,37,35,31)(33,40,42,43)(38,44,41,39);(5,7)(6,10)(8,12)(9,11)(13,16,17,24)(14,19,20,26)(15,23,22,28)(18,25,21,27)(29,33,34,42)(30,38,35,41)(31,39,37,44)(32,40,36,43)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fh,d^2=gi,e^2=fg,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, 4408115);