Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #37683 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2026 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,9)(7,11,12,10)(13,14)(15,24)(16,29)(17,22)(18,21)(19,26)(20,32)(23,30)(25,39)(27,37)(28,34)(31,35)(33,40)(36,44)(38,42)(41,43);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,25)(17,28)(18,26)(20,33)(21,24)(22,37)(23,34)(27,30)(29,43)(31,41)(32,42)(35,39)(36,38)(40,44);(2,4)(6,9)(10,11)(13,16,30,44,18,31,17,32)(14,20,22,35,21,36,23,29)(15,25,27,40,26,41,28,42)(19,33,37,39,24,38,34,43);(5,7,8,12)(6,10,9,11)(14,21)(15,26)(16,31)(22,23)(27,28)(32,44)(33,38)(39,43);(2,4)(13,17,18,30)(14,22,21,23)(15,27,26,28)(16,32,31,44)(19,34,24,37)(20,35,36,29)(25,40,41,42)(33,43,38,39)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592129);