Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(21,22)(23,29)(24,30)(25,32)(26,31)(27,33)(28,34)(35,36);(6,9)(10,12)(13,14)(15,18)(16,17)(19,20)(21,23,26,24,27,35,25,34)(22,28,32,36,33,30,31,29);(2,4)(5,7)(6,10)(8,11)(9,12)(14,17)(18,20)(21,24,27,34)(22,29,33,36)(23,25,35,26)(28,32,30,31);(7,11)(10,12)(13,15,16,19)(14,18,17,20)(21,23,25,34,27,35,26,24)(22,30,31,36,33,28,32,29)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=g,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 382125);