Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393906 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1155 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)(5,6,11,16)(7,17,20,36)(8,21)(9,14)(10,28,29,15)(12,22,32,19)(13,25)(18,35)(23,31)(24,34)(26,30)(27,33);(5,7)(6,12)(8,9)(10,22)(11,20)(13,33)(14,31)(15,36)(16,32)(17,28)(18,24)(19,29)(21,35)(23,27)(25,26)(30,34);(5,8)(6,13)(7,18)(9,22)(10,23)(11,25)(12,30)(14,32)(15,31)(16,21)(17,35)(19,26)(20,27)(24,29)(28,34)(33,36);(2,4)(5,9,10,27,11,26,29,18)(6,14,28,33,16,30,15,35)(7,8,22,23,20,25,19,24)(12,31,17,13,32,34,36,21)Polycyclic presentation:
2^9|a^2=eg,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72250213, #72257140, #72257160 |
| maximal normal subgroup of | #72250213, #72257140, #72257160 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310081);