Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | C2×#3106 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(6,10,14,13)(7,15,17,16)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(1,2,3,4)(5,6,11,10,9,14,8,13)(7,12,15,18,17,20,16,19)(21,23,24,30,25,29,31,35)(22,26,27,33,28,32,34,36);(5,7,9,17)(6,12,14,20)(8,16,11,15)(10,18,13,19);(5,8,9,11)(6,13,14,10)(7,16,17,15)(12,19,20,18)(23,29)(26,32)(30,35)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^b=di,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100805);