Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480476 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1318 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)(7,15,19,28)(8,20)(9,14,16,24)(10,12)(11,18)(13,22)(17,31,33,29)(21,35,30,27)(23,25)(26,36)(32,34);(5,7,16,18)(6,11,24,28)(8,17,30,32)(9,23,10,19)(12,25,14,15)(13,26,35,31)(20,34,27,29)(21,36,22,33);(1,2,3,4)(6,12)(11,25)(13,20)(14,24)(15,28)(26,34)(27,35)(29,31);(5,8,10,22)(6,13,12,20)(7,17,19,33)(9,21,16,30)(11,26,25,34)(14,27,24,35)(15,29,28,31)(18,32,23,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=fi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312602, #72312608, #72312632, … |
| maximal normal subgroup of | #72312602, #72312608, #72312632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402372);