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)(5,6,10,16,11,17,8,14)(7,13,18,27,20,29,21,30)(9,15,22,31,23,12,24,26)(19,28,34,35,33,25,32,36);(6,12)(9,23)(13,25)(14,26)(15,17)(16,31)(19,33)(22,24)(27,35)(28,29)(30,36)(32,34);(2,4)(5,7)(6,13)(8,21)(9,19)(10,18)(11,20)(12,25)(14,30)(15,28)(16,27)(17,29)(22,34)(23,33)(24,32)(26,36)(31,35);(5,8,11,10)(6,14,17,16)(7,18,20,21)(9,22,23,24)(12,26,15,31)(13,27,29,30)(19,32,33,34)(25,35,28,36)Polycyclic presentation:
2^9|a^2=g,d^2=i,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 | #72302925, #72303028 |
| maximal normal subgroup of | #72302925, #72303028 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400633);