Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,12,16,11,17,10,30)(7,18,23,33,22,34,21,13)(8,24,20,35,27,14,26,32)(9,28,19,36,25,31,29,15);(5,7,10,21,11,22,12,23)(6,13,16,18,17,33,30,34)(8,25,26,19,27,9,20,29)(14,15,32,28,24,36,35,31);(1,2,3,4)(5,8,10,26,11,27,12,20)(6,14,30,35,17,24,16,32)(7,19,21,9,22,29,23,25)(13,31,34,36,33,28,18,15);(5,9,10,29,11,25,12,19)(6,15,16,28,17,36,30,31)(7,20,21,8,22,26,23,27)(13,32,18,24,33,35,34,14)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281644);