Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419810 |
| 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×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1151 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)(5,6,11,16,12,17,10,15)(7,13,19,29,21,9,20,28)(8,22,27,14,25,33,26,32)(18,34,36,24,35,31,23,30);(5,7)(6,13)(8,23,25,36)(9,17)(10,20)(11,19)(12,21)(14,31,32,34)(15,28)(16,29)(18,26,35,27)(22,24,33,30);(5,8,10,27,12,25,11,26)(6,14,15,33,17,32,16,22)(7,18,19,23,21,35,20,36)(9,24,28,34,13,30,29,31);(2,4)(5,9,12,13)(6,7,17,21)(8,24,25,30)(10,28,11,29)(14,18,32,35)(15,19,16,20)(22,36,33,23)(26,31,27,34)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=ef,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341256);