Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1933 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(6,11)(7,16)(8,21)(9,12)(13,28)(15,27)(19,20)(22,26)(23,24)(29,35)(31,33)(32,36);(1,2,3,4)(5,6,12,27,10,15,9,11)(7,17,16,13,20,25,19,28)(8,22,23,14,24,26,21,30)(18,29,33,32,34,36,31,35);(5,7,9,16,10,20,12,19)(6,13,11,25,15,28,27,17)(8,18,23,31,24,34,21,33)(14,29,26,35,30,36,22,32);(2,4)(5,8)(6,14)(7,18)(9,23)(10,24)(11,26)(12,21)(13,29)(15,30)(16,31)(17,32)(19,33)(20,34)(22,27)(25,35)(28,36)Polycyclic presentation:
2^9|b^2=ei,c^2=f,e^2=i,f^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,d^b=de,e^c=ei,g^c=gi,e^d=ei,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, 330092);