Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480476 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2029 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)(5,6,9,14,10,12,24,30)(7,15,18,33,19,32,22,35)(8,13,16,26,21,29,23,31)(11,25,28,20,27,34,36,17);(5,7,16,25,10,19,23,34)(6,11,26,33,12,27,31,35)(8,20,24,18,21,17,9,22)(13,15,30,36,29,32,14,28);(2,4);(6,12)(11,27)(13,29)(14,30)(15,32)(26,31)(28,36)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=i,g^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 | #72266731, #72266742, #72266849, … |
| maximal normal subgroup of | #72266731, #72266742, #72266849, … |
| maximal quotient of | #72266739, #72266749, #72266768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401927);