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 | |
1632 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)(5,6,10,9)(7,12,8,11)(13,14)(15,21)(16,24)(17,23)(18,19)(20,25)(22,26)(27,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(6,9)(7,8)(14,19)(16,25)(17,21)(22,28)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(5,7,10,8)(6,11,9,12)(13,15,18,23)(14,17,19,21)(16,22,25,28)(20,26,24,27)(29,30)(31,37)(32,40)(33,36)(34,39)(35,41)(38,42)(43,44);(1,2,3,4)(5,8,10,7)(6,11,9,12)(13,16)(14,20)(15,22)(17,26)(18,25)(19,24)(21,27)(23,28)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,9,10,6)(7,11,8,12)(13,17)(14,15)(16,26)(18,21)(19,23)(20,22)(24,28)(25,27)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=g,d^2=gi,e^2=h,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, 5189950);