Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480460 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35659 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2041 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)(5,6,10,16,11,15,26,27)(7,17,19,9,14,12,25,24)(8,13,21,30,22,29,35,36)(18,32,33,23,31,28,34,20);(5,7,10,25)(6,12,27,9)(8,18,21,34)(11,14,26,19)(13,28,36,23)(15,17,16,24)(20,29,32,30)(22,31,35,33);(2,4)(5,8,11,22)(6,13,15,29)(7,18,14,31)(9,23,24,20)(10,21,26,35)(12,28,17,32)(16,30,27,36)(19,33,25,34);(5,9,11,24)(6,14,15,7)(8,20,22,23)(10,12,26,17)(13,18,29,31)(16,25,27,19)(21,32,35,28)(30,33,36,34)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72302949, #72303269, #72303281, … |
| maximal normal subgroup of | #72302949, #72303269, #72303281, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401898);