Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1620 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)(5,6,10,9)(7,12,8,11)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(30,34)(32,40)(35,37)(38,41);(6,9)(7,8)(15,21)(19,22)(23,27)(26,28)(29,30,33,34)(31,35,39,37)(32,36,40,42)(38,43,41,44);(5,7,10,8)(6,11,9,12)(13,15,18,21)(14,19,16,22)(17,23,25,27)(20,26,24,28)(29,31)(30,35)(32,41)(33,39)(34,37)(36,44)(38,40)(42,43);(1,2)(3,4)(5,8,10,7)(6,11,9,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,30)(31,37)(32,36)(33,34)(35,39)(38,44)(40,42)(41,43);(5,9,10,6)(7,11,8,12)(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)(29,32,33,40)(30,36,34,42)(31,38,39,41)(35,43,37,44)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=h,d^2=h,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5134621);