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 | |
1724 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)(7,10)(8,9)(11,12)(13,14,17,18)(15,19,23,21)(16,24,26,20)(22,28,25,27)(29,30)(31,33)(32,39)(34,35)(36,40)(37,43)(38,44)(41,42);(5,7,9,12)(6,10,8,11)(14,18)(19,21)(20,24)(27,28)(30,35)(31,37)(32,40)(42,44);(6,8)(10,11)(13,15,17,23)(14,19,18,21)(16,25,26,22)(20,28,24,27)(29,31,34,37)(30,33,35,43)(32,41,40,38)(36,44,39,42);(1,2,3,4)(5,8,9,6)(7,11,12,10)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,26)(25,27)(29,32)(30,36)(31,38)(33,42)(34,40)(35,39)(37,41)(43,44);(5,7)(6,10)(8,11)(9,12)(13,16,17,26)(14,20,18,24)(15,22,23,25)(19,27,21,28)(29,33)(30,31)(32,42)(34,43)(35,37)(36,38)(39,41)(40,44)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=f,d^2=fi,e^2=fh,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, 5189560);