Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 | |
1208 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,16,9,15,11,14)(7,12,19,27,18,28,20,32)(8,21,24,31,23,13,25,30)(17,33,22,36,34,26,35,29);(5,7)(6,12)(8,22)(9,18)(10,20)(11,19)(13,29)(14,27)(15,28)(16,32)(17,24)(21,36)(23,35)(25,34)(26,30)(31,33);(1,2,3,4)(5,8,9,23)(6,13,15,21)(7,17,18,34)(10,25,11,24)(12,26,28,33)(14,30,16,31)(19,35,20,22)(27,36,32,29);(6,14,15,16)(7,18)(12,27,28,32)(13,30,21,31)(17,34)(19,20)(22,35)(26,36,33,29)Polycyclic presentation:
2^9|a^2=g,c^2=ef,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285010, #72285470, #72285592 |
| maximal normal subgroup of | #72285010, #72285470, #72285592 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341615);