Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1845 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,9,8)(7,11,12,10)(13,14)(15,19)(16,24)(17,18)(20,26)(21,22)(23,28)(25,27)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,42,44,43);(5,7,9,12)(6,10,8,11)(14,18)(15,21)(20,24)(23,25)(31,38)(33,41)(35,37)(36,40);(5,7,9,12)(6,11,8,10)(13,15,17,21)(14,19,18,22)(16,25,26,23)(20,28,24,27)(29,31,34,38)(30,35,32,37)(33,39,41,44)(36,42,40,43);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,26)(25,27)(29,32,34,30)(31,35,38,37)(33,40,41,36)(39,42,44,43);(5,8,9,6)(7,11,12,10)(13,16)(14,20)(15,23)(17,26)(18,24)(19,27)(21,25)(22,28)(29,33)(30,36)(31,39)(32,40)(34,41)(35,42)(37,43)(38,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=fg,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4948738);