Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
3418 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)(5,6,10,15,11,16,8,13)(7,17,19,34,18,32,25,36)(9,22,21,33,24,14,26,31)(12,27,30,23,28,35,29,20);(5,7)(6,12)(8,19)(9,23)(10,25)(11,18)(13,29)(14,32)(15,30)(16,28)(17,22)(20,24)(21,27)(26,35)(31,36)(33,34);(2,4)(7,18)(12,28)(17,32)(19,25)(20,23)(27,35)(29,30)(34,36);(5,8,11,10)(6,13,16,15)(7,19,18,25)(9,21,24,26)(12,29,28,30)(14,31,22,33)(17,34,32,36)(20,35,23,27)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,c^b=ci,e^b=ei,g^b=ghi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72302901, #72302925, #72303277, … |
| maximal normal subgroup of | #72302901, #72302925, #72303277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400578);