Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33618 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×#472 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1322 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)(5,6,9,14,11,17,12,16)(7,18,20,10,21,27,15,29)(8,22,25,32,26,30,23,13)(19,24,33,36,35,28,31,34);(5,7,10,17,9,15,27,14,11,21,29,6,12,20,18,16)(8,19,28,32,23,33,36,30,26,35,24,13,25,31,34,22);(2,4)(5,8)(6,13)(7,19)(9,23)(10,28)(11,26)(12,25)(14,30)(15,33)(16,22)(17,32)(18,34)(20,31)(21,35)(24,29)(27,36);(5,9,11,12)(6,14,17,16)(7,20,21,15)(8,23,26,25)(10,27,29,18)(13,30,32,22)(19,31,35,33)(24,34,28,36)Polycyclic presentation:
2^9|a^2=g,b^2=eghi,d^2=i,e^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250063, #72250082, #72250107, … |
| maximal normal subgroup of | #72250063, #72250082, #72250107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401857);