Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395693 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#396 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,14,10,15,9,23)(7,16,25,27,19,32,18,12)(8,13,24,28,22,29,21,30)(17,31,34,26,20,35,33,36);(5,7,9,18,10,19,11,25)(6,12,14,16,15,27,23,32)(8,20,21,34,22,17,24,33)(13,26,28,35,29,36,30,31);(5,8,10,22)(6,13,15,29)(7,17,19,20)(9,24,11,21)(12,26,27,36)(14,30,23,28)(16,31,32,35)(18,34,25,33);(2,4)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302890);