Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#3545 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2432 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)(5,6,10,13,11,16,9,15)(7,17,21,34,18,32,20,35)(8,22,25,31,26,14,24,33)(12,27,28,23,29,36,30,19);(5,7)(6,12)(8,23)(9,21)(10,20)(11,18)(13,28)(14,32)(15,30)(16,29)(17,22)(19,26)(24,36)(25,27)(31,34)(33,35);(2,4)(6,13)(9,10)(12,28)(14,33)(15,16)(17,34)(20,21)(22,31)(24,25)(27,36)(29,30)(32,35);(7,18)(12,29)(17,32)(19,23)(20,21)(27,36)(28,30)(34,35)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279356, #72279632, #72280283, … |
| maximal normal subgroup of | #72279356, #72279632, #72280283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400808);