Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(7,11)(10,12)(13,14,18,19)(15,20,22,16)(17,25,26,21)(23,24,28,27)(30,34)(35,37)(36,40)(42,43);(6,9)(10,12)(14,19)(15,22)(17,26)(24,27)(29,30,33,34)(31,37,39,35)(32,40,41,36)(38,42,44,43);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,19)(17,23)(18,22)(21,27)(24,25)(26,28)(29,30)(31,35)(32,36)(33,34)(37,39)(38,42)(40,41)(43,44);(5,7)(6,10)(8,11)(9,12)(13,16,18,20)(14,15,19,22)(17,24,26,27)(21,23,25,28)(29,31)(30,35)(32,38)(33,39)(34,37)(36,42)(40,43)(41,44);(13,17,18,26)(14,21,19,25)(15,23,22,28)(16,24,20,27)(29,32,33,41)(30,36,34,40)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fgi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 514171);