Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | |
1653 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)(5,6,10,16,12,9,11,15)(7,17,21,19,22,32,20,13)(8,14,25,31,27,24,26,30)(18,29,23,33,35,36,34,28);(5,7,10,21,12,22,11,20)(6,13,16,17,9,19,15,32)(8,23,25,35,27,34,26,18)(14,29,31,33,24,36,30,28);(1,2,3,4)(5,8,11,26,12,27,10,25)(6,14,15,30,9,24,16,31)(7,18,21,23,22,35,20,34)(13,28,17,29,19,33,32,36);(5,9)(6,12)(7,19)(8,24)(10,15)(11,16)(13,22)(14,27)(17,20)(18,33)(21,32)(23,36)(25,30)(26,31)(28,35)(29,34)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276041);