Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32894 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3571 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,18)(8,14)(9,16)(10,15)(11,17)(12,20)(13,22)(19,26)(21,31)(23,30)(24,28)(25,29)(27,34)(32,35)(33,36);(1,3,4,2)(5,7)(6,12)(8,19)(9,24)(10,20)(11,21)(13,26)(14,27)(15,31)(16,18)(17,28)(22,35)(23,36)(25,34)(29,33)(30,32);(5,8)(6,13)(7,19)(9,22)(10,25)(11,23)(12,26)(14,15)(16,30)(17,29)(18,32)(20,34)(21,36)(24,35)(27,31)(28,33);(5,8)(6,14)(7,19)(9,25)(10,22)(11,23)(12,27)(13,15)(16,29)(17,30)(18,33)(20,35)(21,36)(24,34)(26,31)(28,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,g^b=gi,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280888, #72281084, #72281232, … |
| maximal normal subgroup of | #72280888, #72281084, #72281232, … |
| maximal quotient of | #72281637 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299791);