Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,16,11,17,10,26)(7,18,21,13,22,32,27,31)(8,23,20,33,24,14,28,34)(9,15,29,35,25,30,19,36);(1,2)(3,4)(5,7,12,21,11,22,10,27)(6,13,16,32,17,31,26,18)(8,19,20,9,24,29,28,25)(14,15,34,35,23,30,33,36);(2,4)(5,8,10,20,11,24,12,28)(6,14,16,33,17,23,26,34)(7,19,21,25,22,29,27,9)(13,30,18,36,31,15,32,35);(5,9,10,19,11,25,12,29)(6,15,16,35,17,30,26,36)(7,20,21,24,22,28,27,8)(13,23,18,34,31,14,32,33)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=e,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329911);