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 | |
1602 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)(7,11)(8,10)(13,14,18,15)(16,21,24,19)(17,22,26,20)(23,27,25,28)(29,30)(31,37)(32,34)(33,40)(35,38)(36,41)(39,42)(43,44);(5,6,9,12)(7,8,11,10)(14,15)(19,21)(20,22)(27,28)(30,32)(31,38)(36,40)(39,44);(5,7,9,11)(6,10,12,8)(13,15,18,14)(16,21,24,19)(17,22,26,20)(23,28,25,27)(29,31)(30,35)(32,37)(33,39)(34,38)(36,42)(40,43)(41,44);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27)(29,32,34,30)(31,37,38,35)(33,36,41,40)(39,42,44,43);(5,8,9,10)(6,11,12,7)(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)(29,33,34,41)(30,36,32,40)(31,39,38,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=fg,d^2=f,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, 4773138);