Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×H(5,𝔽2) | C23 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,12)(9,13)(10,14)(11,18)(16,30)(17,31)(19,35)(20,27)(21,28)(22,36)(23,29)(24,32)(25,33)(26,34);(5,7,10,17)(6,11,14,25)(8,19,21,16)(9,18,23,33)(12,26,28,24)(13,15,29,31)(20,34,22,32)(27,35,36,30);(5,8,10,21)(6,12,14,28)(7,16,17,19)(9,22,23,20)(11,24,25,26)(13,27,29,36)(15,30,31,35)(18,34,33,32);(1,3,4,2)(7,17)(11,25)(15,31)(16,19)(18,33)(24,26)(30,35)(32,34)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=g,b^a=be,c^a=cf,h^a=hi,c^b=ci,d^b=di,f^b=fhi,e^c=eh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281265 |
| maximal normal subgroup of | #72281265 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402974);