Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1630 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)(5,6,9,7)(8,10,12,11)(13,14)(15,19)(16,18)(17,20)(21,26)(22,23)(24,25)(27,28)(29,30,33,36)(31,37,39,34)(32,40,42,35)(38,43,41,44);(5,7,9,6)(8,10,12,11)(14,16)(19,23)(20,24)(26,27)(29,30)(31,34)(32,35)(33,36)(37,39)(38,43)(40,42)(41,44);(5,8,9,12)(6,10,7,11)(13,15,18,22)(14,19,16,23)(17,21,25,28)(20,26,24,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,44)(40,42)(41,43);(1,2)(3,4)(13,16,18,14)(15,19,22,23)(17,20,25,24)(21,27,28,26)(29,31,33,39)(30,34,36,37)(32,41,42,38)(35,44,40,43);(13,17)(14,20)(15,21)(16,24)(18,25)(19,26)(22,28)(23,27)(29,32,33,42)(30,35,36,40)(31,38,39,41)(34,43,37,44)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=fgh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2644421);