Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C22×Q8 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1278 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)(7,17)(8,14)(9,23)(10,16)(11,13)(12,19)(15,24)(18,32)(20,35)(21,31)(22,30)(25,29)(26,34)(27,33)(28,36);(5,7,10,20,11,21,8,18)(6,12,16,28,13,26,14,27)(9,19,22,33,24,34,25,36)(15,17,29,32,23,31,30,35);(1,3,4,2)(6,13)(12,26)(14,16)(15,23)(17,31)(27,28)(29,30)(32,35);(5,8,11,10)(6,14,13,16)(7,18,21,20)(9,22,24,25)(12,27,26,28)(15,29,23,30)(17,32,31,35)(19,33,34,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=i,g^2=i,b^a=be,c^a=ci,e^a=ei,g^a=ghi,e^b=eh,h^b=hi,g^e=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72293121, #72293122, #72293126, … |
| maximal normal subgroup of | #72293121, #72293122, #72293126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400214);