Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437428 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3081 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)(7,10)(8,9)(11,12)(13,14,19,24)(15,25,28,38)(16,23,18,21)(17,22,31,37)(20,32,34,26)(27,39,42,43)(29,36,30,35)(33,41,44,40);(1,2)(3,4)(13,15,19,28)(14,20,24,34)(16,26,18,32)(17,27,31,42)(21,25,23,38)(22,33,37,44)(29,40,30,41)(35,39,36,43);(2,4)(6,9)(10,12)(13,16,19,18)(14,21,24,23)(15,26,28,32)(17,30,31,29)(20,25,34,38)(22,36,37,35)(27,41,42,40)(33,43,44,39);(2,4)(13,17)(14,22)(15,27)(16,29)(18,30)(19,31)(20,33)(21,35)(23,36)(24,37)(25,39)(26,40)(28,42)(32,41)(34,44)(38,43);(5,7,8,11)(6,10,9,12)(17,31)(22,37)(27,42)(29,30)(33,44)(35,36)(39,43)(40,41)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360990);