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 |
| prime radical | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,11,8,10)(13,14,19,25)(15,27,29,21)(16,31,34,43)(17,36,37,44)(18,26,20,24)(22,35,40,33)(23,32,39,38)(28,41,30,42);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,32)(17,35)(18,28)(19,29)(20,30)(22,36)(23,31)(24,41)(25,27)(26,42)(33,37)(34,38)(39,43)(40,44);(5,7,9,8)(6,11,12,10)(13,16,19,34)(14,22,25,40)(15,17,29,37)(18,35,20,33)(21,39,27,23)(24,43,26,31)(28,38,30,32)(36,41,44,42);(2,4)(13,17,19,37)(14,23,25,39)(15,16,29,34)(18,32,20,38)(21,40,27,22)(24,44,26,36)(28,33,30,35)(31,41,43,42);(5,8,9,7)(6,11,12,10)(13,18,19,20)(14,24,25,26)(15,28,29,30)(16,33,34,35)(17,32,37,38)(21,41,27,42)(22,31,40,43)(23,44,39,36)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=f,d^2=i,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,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, 7427965);