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×#3499 | 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 | |
3280 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)(5,6,10,15,11,16,7,17)(8,13,22,29,20,27,18,31)(9,14,19,26,24,30,23,12)(21,28,32,35,33,36,34,25);(5,7,11,10)(6,12,16,26)(8,18,20,22)(9,23,24,19)(13,25,27,35)(14,15,30,17)(21,34,33,32)(28,29,36,31);(2,4)(5,8)(6,13)(7,18)(9,21)(10,22)(11,20)(12,25)(14,28)(15,29)(16,27)(17,31)(19,32)(23,34)(24,33)(26,35)(30,36);(8,20)(13,27)(18,22)(21,33)(25,35)(28,36)(29,31)(32,34)Polycyclic presentation:
2^9|a^2=g,b^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 | #72302902, #72302925, #72303181, … |
| maximal normal subgroup of | #72302902, #72302925, #72303181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400635);