Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,16)(15,19,20,18)(21,22,25,27)(23,26,31,34)(24,28,32,36)(29,33,30,35);(13,15,17,20)(14,18,16,19)(21,23)(22,26)(24,29)(25,31)(27,34)(28,33)(30,32)(35,36);(1,2,3,4)(6,9)(10,12)(14,16)(18,19)(23,29)(24,32)(26,33)(28,36)(30,31)(34,35);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,19,20,18)(22,27)(26,34)(28,36)(33,35)Polycyclic presentation:
2^9|a^2=ei,b^2=e,c^2=h,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,d^c=de,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290094);