Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1786 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)(13,14,19,24)(15,20,28,37)(16,29,32,39)(17,23,18,21)(22,33,40,31)(25,36,27,35)(26,38,42,44)(30,34,41,43);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,30)(17,27)(18,25)(19,28)(21,36)(22,38)(23,35)(24,37)(26,31)(29,34)(32,41)(33,42)(39,43)(40,44);(14,21)(15,25)(17,18)(22,29)(23,24)(26,30)(27,28)(31,33)(35,36)(38,44)(39,40)(41,42);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,22)(15,26)(17,31)(18,33)(19,32)(20,34)(21,29)(23,39)(24,40)(25,30)(27,41)(28,42)(35,38)(36,44)(37,43);(5,8,9,7)(6,11,12,10)(13,17,19,18)(14,23,24,21)(15,27,28,25)(16,31,32,33)(20,35,37,36)(22,39,40,29)(26,41,42,30)(34,38,43,44)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=f,e^2=fi,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=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7430404);