Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4462490 |
| automorphism | Aut(G) order 4194304 |
| 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 | |
1758 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)(13,14,18,21)(15,22,19,16)(17,25,26,20)(23,24,27,28)(29,30)(31,37)(32,40)(33,36)(34,39)(35,41)(38,42)(43,44);(7,11)(10,12)(15,19)(16,22)(23,27)(24,28)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(5,6,8,9)(7,10,11,12)(13,15,18,19)(14,16,21,22)(17,23,26,27)(20,24,25,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,19)(15,21)(17,24)(18,22)(20,27)(23,25)(26,28)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(13,17,18,26)(14,20,21,25)(15,23,19,27)(16,24,22,28)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=g,d^2=gi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4371358);