Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2946 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(10,12)(13,14)(15,25)(16,30)(17,24)(18,23)(19,22)(20,28)(21,33)(26,38)(27,37)(29,36)(31,35)(32,39)(34,40)(41,43)(42,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,31)(17,29)(18,28)(19,27)(21,38)(22,36)(23,37)(24,25)(26,34)(30,41)(32,42)(33,43)(35,39)(40,44);(6,9)(10,12)(13,16,19,34)(14,21,22,39)(15,26,27,31)(17,33,18,32)(20,35,36,38)(23,30,24,40)(25,41,37,44)(28,43,29,42);(5,7,8,11)(6,10,9,12)(14,22)(20,36)(21,39)(23,24)(25,37)(30,40)(35,38)(41,44);(15,27)(20,36)(25,37)(26,31)(28,29)(35,38)(41,44)(42,43)Polycyclic presentation:
2^9|a^2=fg,c^2=i,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342644);