Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,11)(7,12,8,10)(13,14,16,18)(15,19,20,17)(21,22,25,27)(23,29,31,36)(24,32,33,28)(26,34,35,30);(5,7,9,8)(6,10,11,12)(23,30)(24,33)(26,29)(28,32)(31,34)(35,36);(1,2,3,4)(5,8,9,7)(6,10,11,12)(21,23,24,30,25,31,33,34)(22,26,28,29,27,35,32,36);(13,15,16,20)(14,17,18,19)(22,27)(26,35)(28,32)(29,36)Polycyclic presentation:
2^9|a^2=eh,b^2=e,c^2=ef,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322553);