Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456658 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| 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 | |
925 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)(7,16,20,17)(8,10)(9,22)(11,15)(12,19,28,21)(13,23)(14,26)(18,27,35,33)(24,29)(25,30)(31,34,32,36);(5,7,11,20)(6,12)(8,16)(9,18,24,35)(10,19,26,21)(13,27)(14,17)(15,28)(22,31)(23,34,25,36)(29,32)(30,33);(5,8,10,15,11,14,26,6)(7,17,19,12,20,16,21,28)(9,13,23,29,24,30,25,22)(18,33,34,31,35,27,36,32);(2,4)(5,9,11,24)(6,13,15,30)(7,18,20,35)(8,22,14,29)(10,25,26,23)(12,27,28,33)(16,31,17,32)(19,36,21,34)Polycyclic presentation:
2^9|a^2=g,b^2=ef,c^2=e,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,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, 367910);