Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454041 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3020 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(6,12)(7,17)(8,22)(9,10)(13,18)(14,30)(15,16)(19,34)(20,21)(24,25)(28,36)(31,33);(5,6,11,16)(7,18,21,13)(8,14,25,30)(9,15,10,12)(17,26,20,29)(19,33,34,28)(22,32,24,27)(23,31,35,36);(2,4)(5,7)(6,13)(8,23)(9,17)(10,20)(11,21)(12,26)(14,31)(15,29)(16,18)(19,24)(22,34)(25,35)(27,33)(28,32)(30,36);(5,8)(6,14)(7,19)(9,22)(10,24)(11,25)(12,27)(13,28)(15,32)(16,30)(17,23)(18,33)(20,35)(21,34)(26,31)(29,36)Polycyclic presentation:
2^9|a^2=ef,b^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361885);