Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #10991 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1687 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)(3,11,15,23)(4,16,18,8)(5,20,21,9)(6,10)(7,13)(12,25,27,31)(14,28)(17,22)(19,24)(26,29)(30,32);(1,3)(2,7)(4,17)(5,14)(6,15)(8,23)(9,12)(10,22)(11,16)(13,18)(19,31)(20,27)(21,29)(24,28)(25,32)(26,30);(1,2)(3,12)(4,8)(5,9)(6,10)(7,14)(11,25)(13,26)(15,27)(16,18)(17,28)(19,24)(20,21)(22,29)(23,31)(30,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72210522, C2×#147761, #72267994, … |
| maximal normal subgroup of | #72210522, C2×#147761, #72267994, … |
| maximal quotient of | C2×#147761 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54957);