Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476112 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,16)(8,12)(9,13)(10,15)(11,19)(14,24)(17,32)(18,26)(20,27)(21,29)(22,31)(23,30)(25,34)(28,33)(35,36);(1,3,4,2)(5,7)(6,11)(8,17)(9,18)(10,20)(12,28)(13,16)(14,27)(15,26)(19,24)(21,33)(22,34)(23,35)(25,30)(29,36)(31,32);(2,3)(5,8,10,23)(6,12,15,30)(7,17,20,35)(9,21,24,22)(11,25,26,28)(13,29,14,31)(16,32,27,36)(18,33,19,34);(2,3)(6,13)(7,18)(8,21)(9,24)(14,15)(16,27)(19,20)(22,23)(25,32)(28,36)(29,31)(33,34)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,e^c=ei,h^c=hi,e^d=ei,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384189);