Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161262 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3514 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,15,10,14,8,13)(7,12,16,20,17,11,18,19)(21,22)(23,29,31,36)(24,28)(25,27)(26,30,34,32)(33,35);(5,7)(6,11)(8,16)(9,18)(10,17)(12,14)(13,19)(15,20)(21,23)(22,26)(24,32)(25,31)(27,34)(28,36)(29,35)(30,33);(2,4)(5,7)(6,12)(8,18)(9,16)(10,17)(11,14)(13,19)(15,20);(5,8,10,9)(6,13,14,15)(7,16,17,18)(11,19,12,20)(22,27)(26,34)(28,35)(29,36)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=ghi,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92340);