Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421159 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| 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 | |
983 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)(5,6,10,23,9,11,22,14)(7,15,16,28,18,12,24,26)(8,19,21,29,20,13,25,30)(17,27,33,36,34,31,32,35);(2,4)(6,11)(7,16,18,24)(8,20)(12,26,15,28)(14,23)(17,32,34,33)(21,25)(27,35,31,36);(1,2,3,4)(5,7)(6,12,11,15)(8,17)(9,18)(10,24)(13,27,19,31)(14,26,23,28)(16,22)(20,34)(21,32)(25,33)(29,36,30,35);(5,8,9,20)(6,13,11,19)(7,17,18,34)(10,25,22,21)(12,27,15,31)(14,30,23,29)(16,32,24,33)(26,35,28,36)Polycyclic presentation:
2^9|a^2=efg,b^2=ei,c^2=ef,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342035);