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 | |
1885 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)(7,15)(8,20)(9,24)(12,28)(14,27)(18,19)(21,26)(22,23)(29,35)(31,33)(32,36);(1,2,3,4)(5,6,9,11,10,14,24,27)(7,16,18,28,19,25,15,12)(8,21,20,30,23,26,22,13)(17,29,31,35,34,36,33,32);(5,7,9,15,10,19,24,18)(6,12,11,25,14,28,27,16)(8,17,22,31,23,34,20,33)(13,29,26,35,30,36,21,32);(2,4)(5,8)(6,13)(7,17)(9,22)(10,23)(11,26)(12,29)(14,30)(15,31)(16,32)(18,33)(19,34)(20,24)(21,27)(25,35)(28,36)Polycyclic presentation:
2^9|b^2=e,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, 330091);