Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455962 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,12,17,10,18,11,16)(7,13,22,30,21,32,28,31)(8,23,26,14,15,9,25,27)(19,33,24,36,34,35,29,20);(5,7)(6,13)(8,24)(9,20)(10,21)(11,22)(12,28)(14,33)(15,29)(16,30)(17,31)(18,32)(19,26)(23,36)(25,34)(27,35);(5,8,10,15)(6,14,18,27)(7,19,21,34)(9,17,23,16)(11,26,12,25)(13,20,32,36)(22,29,28,24)(30,35,31,33);(2,4)(5,9,11,14,10,23,12,27)(6,15,16,26,18,8,17,25)(7,20,22,33,21,36,28,35)(13,29,30,19,32,24,31,34)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369519);