Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419895 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,12,19,28)(8,21,23,32)(9,15,10,14)(13,25,31,24)(17,29,20,26)(18,34,22,36)(27,33,30,35);(1,2)(3,4)(5,7)(6,12)(8,22)(9,17)(10,20)(11,19)(13,30)(14,26)(15,29)(16,28)(18,23)(21,34)(24,35)(25,33)(27,31)(32,36);(2,4)(7,17)(9,10)(12,26)(14,15)(18,33)(19,20)(21,32)(22,35)(24,25)(27,36)(28,29)(30,34);(5,8)(6,13)(7,18,19,22)(9,24)(10,25)(11,23)(12,27,28,30)(14,21)(15,32)(16,31)(17,33,20,35)(26,36,29,34)Polycyclic presentation:
2^9|a^2=i,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285470, #72285534, #72285571 |
| maximal normal subgroup of | #72285470, #72285534, #72285571 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335022);