Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455962 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,14,10,15,9,16)(7,17,19,12,20,29,24,27)(8,21,25,28,23,30,22,13)(18,26,32,35,33,36,34,31);(1,2,3,4)(5,7,11,19,10,20,9,24)(6,12,14,29,15,27,16,17)(8,18,25,32,23,33,22,34)(13,26,21,35,28,36,30,31);(2,4)(8,22)(9,11)(13,21)(14,16)(17,29)(18,32)(19,24)(23,25)(26,31)(28,30)(33,34)(35,36);(5,8,9,22,10,23,11,25)(6,13,16,30,15,28,14,21)(7,18,24,34,20,33,19,32)(12,26,17,31,27,36,29,35)Polycyclic presentation:
2^9|a^2=ef,b^2=fgh,d^2=efi,e^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,g^b=gi,d^c=de,e^c=ei,h^c=hi,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, 380666);