Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14,16,18)(15,17,19,20)(21,22,24,28)(23,30,31,26)(25,27,33,29)(32,35,36,34);(2,3)(5,7,8,12)(6,9,10,11)(13,15)(14,17)(16,19)(18,20);(6,10)(9,11)(15,19)(17,20)(21,23,24,31)(22,26,28,30)(25,32,33,36)(27,34,29,35);(1,3,4,2)(7,12)(9,11)(22,27)(25,33)(26,34)(28,29)(30,35)(32,36)Polycyclic presentation:
2^9|a^2=efg,b^2=fi,c^2=i,d^2=efi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344154);