Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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)(6,9)(7,11)(13,14,18,19)(15,20,24,27)(16,17,26,21)(22,28,25,23)(29,30,33,36)(31,37,39,34)(32,35,41,44)(38,43,40,42);(14,19)(17,21)(20,27)(23,28)(29,31,33,39)(30,34,36,37)(32,40,41,38)(35,43,44,42);(5,6,8,9)(7,12,11,10)(13,15)(14,20)(16,25)(17,23)(18,24)(19,27)(21,28)(22,26)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,41)(43,44);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,22)(17,19)(18,26)(20,23)(24,25)(27,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,43)(41,44);(13,17,18,21)(14,16,19,26)(15,23,24,28)(20,25,27,22)(29,32,33,41)(30,35,36,44)(31,38,39,40)(34,42,37,43)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=gh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4843562);