Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33557 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1889 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,4,9,13,22)(3,12,16,28,14,7,21,25)(5,10,19,27,17,8,20,26)(15,23,29,31,18,24,30,32);(1,3,13,21,4,14,6,16)(2,7,22,28,9,12,11,25)(5,18,19,30,17,15,20,29)(8,23,26,31,10,24,27,32);(2,8)(5,17)(7,23)(9,10)(11,27)(12,24)(15,18)(19,20)(22,26)(25,32)(28,31)(29,30);(3,14)(7,12)(15,18)(16,21)(23,24)(25,28)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fhi,h^a=hi,d^b=di,f^b=fhi,e^c=eh,f^c=fi,g^c=gh,f^e=fi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236738, #72236884, #72236949, … |
| maximal normal subgroup of | #72236884, #72236949, #72237532, … |
| maximal quotient of | #72267987, #72279970, #72312778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401801);