Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | #1999 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
765 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)(6,10)(8,11)(12,16)(13,17)(15,18)(19,20)(21,22,25,29,26,30,32,28)(23,31,24,27,33,36,34,35);(5,6,11,17,9,13,8,10)(7,12,18,20,14,19,15,16)(21,23,32,34,26,33,25,24)(22,27,29,36,30,35,28,31);(2,4)(5,7)(6,12)(8,15)(9,14)(10,16)(11,18)(13,19)(17,20)(23,33)(24,34)(27,35)(31,36);(7,14)(12,19)(15,18)(16,20)(21,24,25,33,26,34,32,23)(22,27,29,36,30,35,28,31)Polycyclic presentation:
2^9|a^2=g,b^2=eh,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,d^c=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 118076);