Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×D4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
3382 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)(5,6,9,14,10,15,22,29)(7,12,18,27,16,25,31,35)(8,13,20,28,19,11,21,24)(17,26,32,36,30,23,33,34);(6,11)(8,19)(12,23)(13,15)(14,28)(17,30)(20,21)(24,29)(25,26)(27,36)(32,33)(34,35);(2,4)(5,7)(6,12)(8,17)(9,18)(10,16)(11,23)(13,26)(14,27)(15,25)(19,30)(20,32)(21,33)(22,31)(24,34)(28,36)(29,35);(7,16)(12,25)(17,30)(18,31)(23,26)(27,35)(32,33)(34,36)Polycyclic presentation:
2^9|a^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=ghi,d^c=di,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72302905, #72302925, #72303186, … |
| maximal normal subgroup of | #72302905, #72302925, #72303186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400632);