Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1610 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)(5,6,9,8)(7,10,11,12)(13,14)(15,20)(16,22)(17,25)(18,19)(21,26)(23,24)(27,28)(30,34)(31,37)(36,40)(39,43);(7,11)(10,12)(14,19)(15,22)(17,26)(24,27)(29,30,33,34)(31,38,37,35)(32,40,41,36)(39,42,43,44);(5,7,9,11)(6,10,8,12)(13,15,18,22)(14,20,19,16)(17,23,26,28)(21,27,25,24)(29,31,33,37)(30,35,34,38)(32,39,41,43)(36,42,40,44);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,16,18,20)(14,15,19,22)(17,27,26,24)(21,28,25,23)(29,30)(31,38)(32,40)(33,34)(35,37)(36,41)(39,42)(43,44);(13,17,18,26)(14,21,19,25)(15,23,22,28)(16,24,20,27)(29,32,33,41)(30,36,34,40)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=h,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2983226);