Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| 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 | #2580 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,18,21,32)(8,13)(9,14,27,33)(10,15)(11,17)(12,20,31,23)(16,28)(19,35)(22,36,26,29)(24,34)(25,30);(1,2)(3,4)(5,7,10,23,11,21,28,20)(6,12,15,32,17,31,16,18)(8,22,24,27,25,26,19,9)(13,33,34,29,30,14,35,36);(2,4)(5,8,10,24,11,25,28,19)(6,13,15,34,17,30,16,35)(7,9,23,22,21,27,20,26)(12,29,32,14,31,36,18,33);(5,9)(6,14)(7,19)(8,23)(10,22)(11,27)(12,30)(13,31)(15,36)(16,29)(17,33)(18,34)(20,25)(21,24)(26,28)(32,35)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322946 |
| maximal normal subgroup of | #72322946 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 333945);