Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#3199 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,20,21,12)(9,23)(10,14)(11,17,26,32)(13,24)(16,29)(18,34)(19,30)(22,28,36,35)(25,33,27,31);(5,7)(6,11)(8,17)(9,16)(10,19)(12,15)(13,25)(14,26)(18,24)(20,30)(21,32)(22,31)(23,27)(28,34)(29,35)(33,36);(2,4)(7,16)(9,24)(11,25)(12,20)(13,23)(15,29)(17,31)(18,19)(22,36)(26,27)(30,34)(32,33)Polycyclic presentation:
2^9|a^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,d^c=dh,f^c=fi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72311943, #72311949, #72311969, … |
| maximal normal subgroup of | #72311943, #72311949, #72311969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55033);