Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #147752 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10991 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#3027 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,7,17,20)(3,11,13,25,14,24,27,32)(4,9,16,23,15,19,28,31)(8,21,22,26,18,29,30,12);(2,7)(4,15)(8,18)(9,19)(13,27)(16,28)(21,29)(22,30);(1,3)(2,8)(4,12)(5,13)(6,14)(7,18)(9,11)(10,22)(15,26)(16,21)(17,27)(19,24)(20,30)(23,25)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,g^b=gh,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72267969, #72267994, #72267995, … |
| maximal normal subgroup of | #72267969, #72267994, #72267995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54949);