Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456417 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| 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 | |
1223 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)(3,4)(5,6,11,14,10,15,9,26)(7,16,18,12,20,30,19,29)(8,21,24,32,25,13,27,23)(17,33,22,36,35,31,34,28);(5,7,11,18,10,20,9,19)(6,12,14,30,15,29,26,16)(8,22,24,35,25,34,27,17)(13,31,23,28,21,33,32,36);(5,8,9,24,10,25,11,27)(6,13,14,32,15,21,26,23)(7,17,18,34,20,35,19,22)(12,28,16,33,29,36,30,31);(2,4)(5,6,10,15)(7,12,20,29)(8,23,25,32)(9,26,11,14)(13,24,21,27)(16,18,30,19)(17,31,35,33)(22,28,34,36)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=e,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,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, 364133);