Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#390 |
| prime radical | C2×#3002 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2329 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)(5,6,10,14,11,15,9,23)(7,12,17,28,18,29,24,27)(8,19,21,13,22,31,25,30)(16,32,34,36,20,35,33,26);(2,4)(5,7)(6,12)(8,20,22,16)(9,24)(10,17)(11,18)(13,26,30,36)(14,28)(15,29)(19,35,31,32)(21,33,25,34)(23,27);(5,8)(6,13)(7,16)(9,25)(10,21)(11,22)(12,26)(14,31)(15,30)(17,34)(18,20)(19,23)(24,33)(27,35)(28,32)(29,36);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273800);