Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1616 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)(13,14,18,16)(15,21,22,19)(17,24,25,20)(23,26,28,27)(30,31)(32,39)(33,41)(43,44);(6,9)(10,12)(15,22)(17,25)(19,21)(20,24)(29,30,34,31)(32,35,39,37)(33,38,41,36)(40,44,42,43);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,21)(17,23)(18,22)(20,26)(24,27)(25,28)(29,31,34,30)(32,35,39,37)(33,36,41,38)(40,44,42,43);(1,2)(3,4)(13,16,18,14)(15,19,22,21)(17,24,25,20)(23,26,28,27)(29,32)(30,35)(31,37)(33,42)(34,39)(36,44)(38,43)(40,41);(5,7,8,11)(6,10,9,12)(13,17,18,25)(14,20,16,24)(15,23,22,28)(19,26,21,27)(29,33,34,41)(30,36,31,38)(32,40,39,42)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=gh,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4057844);