Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,10)(7,11,8,12)(13,14,19,24)(15,20,27,40)(16,29,26,42)(17,32,25,44)(18,34,37,23)(21,33,39,31)(22,36,38,30)(28,43,35,41);(1,2)(3,4)(6,10)(11,12)(13,15,19,27)(14,20,24,40)(16,30,26,36)(17,33,25,31)(18,35,37,28)(21,32,39,44)(22,42,38,29)(23,43,34,41);(13,16,19,26)(14,21,24,39)(15,25,27,17)(18,33,37,31)(20,38,40,22)(23,42,34,29)(28,30,35,36)(32,43,44,41);(2,4)(5,7,9,8)(6,11,10,12)(13,17,19,25)(14,22,24,38)(15,26,27,16)(18,36,37,30)(20,39,40,21)(23,44,34,32)(28,31,35,33)(29,43,42,41);(5,8,9,7)(6,11,10,12)(15,27)(16,26)(20,40)(21,39)(28,35)(29,42)(31,33)(41,43)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429281);