Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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×#3035 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1481 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,11)(8,12,20,30)(9,14)(10,22)(13,15)(16,33,34,25)(17,31)(18,26)(19,28,35,36)(21,32,24,29)(23,27);(1,2,4,3)(5,7,15,17,9,18,10,23)(6,11,13,31,14,26,22,27)(8,19,21,16,20,35,24,34)(12,28,32,33,30,36,29,25);(5,8)(6,12)(7,16)(9,20)(10,24)(11,25)(13,32)(14,30)(15,21)(17,35)(18,34)(19,23)(22,29)(26,33)(27,36)(28,31);(1,3,4,2)Polycyclic presentation:
2^9|a^2=h,b^2=efgh,d^2=e,g^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,g^c=gi,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274216);