Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(14,17)(18,19)(21,22,25,30)(23,27,31,35)(24,28,26,29)(32,36,33,34);(6,9)(10,12)(13,14,16,17)(15,19,20,18)(21,23)(22,27)(24,32)(25,31)(26,33)(28,36)(29,34)(30,35);(2,4)(7,11)(10,12)(13,15)(14,18)(16,20)(17,19)(22,28)(23,31)(24,26)(27,34)(29,30)(35,36);(5,7,8,11)(6,10,9,12)(15,20)(18,19)(22,29)(24,26)(27,34)(28,30)(32,33)(35,36)Polycyclic presentation:
2^9|a^2=i,b^2=gi,d^2=egi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 377470);