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 | |
1722 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)(7,8)(9,10)(11,12)(13,14,18,19)(15,17,21,27)(16,23,24,20)(22,28,25,26)(29,30,34,37)(31,35,32,39)(33,36,41,44)(38,42,40,43);(6,10)(7,12)(14,19)(15,21)(16,24)(26,28)(29,31,34,32)(30,35,37,39)(33,40,41,38)(36,43,44,42);(7,12)(8,11)(13,15)(14,17)(16,25)(18,21)(19,27)(20,28)(22,24)(23,26)(29,32,34,31)(30,35,37,39)(33,40,41,38)(36,42,44,43);(1,2,3,4)(5,7,9,12)(6,11,10,8)(13,16,18,24)(14,20,19,23)(15,22,21,25)(17,26,27,28)(29,30)(31,35)(32,39)(33,36)(34,37)(38,42)(40,43)(41,44);(5,8,9,11)(6,12,10,7)(13,17)(14,21)(15,19)(16,26)(18,27)(20,25)(22,23)(24,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,44)(39,43)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=gi,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, 5189963);