Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460861 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
617 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)(5,6,10,15,9,12,11,24)(7,16,19,25,20,27,17,13)(8,14,22,29,21,26,23,30)(18,33,32,36,34,31,35,28);(6,12)(7,17,20,19)(13,27,25,16)(14,26)(15,24)(18,32,34,35)(28,33,36,31)(29,30);(5,7,11,19,9,20,10,17)(6,13,24,16,12,25,15,27)(8,18,23,32,21,34,22,35)(14,28,30,33,26,36,29,31);(2,4)(5,8,10,23,9,21,11,22)(6,14,24,29,12,26,15,30)(7,18,17,32,20,34,19,35)(13,28,16,31,25,36,27,33)Polycyclic presentation:
2^9|a^2=f,b^2=ei,c^2=ef,d^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377159);