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×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,13,11,16,9,15)(7,12,17,28,20,25,19,27)(8,21,24,30,22,29,23,14)(18,32,33,26,34,35,31,36);(1,2,3,4)(5,7,11,20)(6,12,16,25)(8,18,22,34)(9,19,10,17)(13,28,15,27)(14,26,30,36)(21,35,29,32)(23,31,24,33);(2,4)(6,13)(7,17)(8,22)(9,10)(12,25)(14,21)(15,16)(18,33)(19,20)(29,30)(31,34)(32,35);(5,8,9,24,11,22,10,23)(6,14,15,21,16,30,13,29)(7,18,19,33,20,34,17,31)(12,26,27,35,25,36,28,32)Polycyclic presentation:
2^9|a^2=fi,b^2=egh,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380332);