Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1682 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)(5,6,8,9)(7,11,12,10)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(29,30,34,37)(31,35,39,32)(33,41,43,36)(38,40,44,42);(6,9)(10,11)(15,21)(18,25)(22,27)(26,28)(29,31,34,39)(30,32,37,35)(33,38,43,44)(36,40,41,42);(5,6)(7,10)(8,9)(11,12)(13,15,17,21)(14,18,20,25)(16,22,24,27)(19,26,23,28)(29,31)(30,35)(32,37)(33,38)(34,39)(36,42)(40,41)(43,44);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,32,34,35)(30,31,37,39)(33,42,43,40)(36,44,41,38);(5,7,8,12)(6,10,9,11)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,33,34,43)(30,36,37,41)(31,38,39,44)(32,40,35,42)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=fh,d^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4773578);