Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| 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 | |
764 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)(12,17)(13,18)(14,20)(16,19)(21,22,25,30,26,31,33,29)(23,32,24,27,34,36,35,28);(5,6,11,18,9,13,8,10)(7,14,19,17,15,12,16,20)(21,23,33,35,26,34,25,24)(22,27,30,36,31,28,29,32);(2,4)(5,7)(6,12)(8,16)(9,15)(10,17)(11,19)(13,14)(18,20);(7,15)(12,14)(16,19)(17,20)(21,24,25,34,26,35,33,23)(22,28,30,32,31,27,29,36)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,d^a=di,e^a=eh,c^b=chi,d^c=dhi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 118186);