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 | #60387 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1667 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,8,14,6,12,23,30)(3,15,17,21,18,20,27,26)(4,19,22,16,9,10,28,29)(5,11,7,13,24,31,25,32);(1,3)(2,9)(4,12)(5,16)(6,18)(7,19)(8,27)(10,25)(11,15)(13,21)(14,28)(17,23)(20,31)(22,30)(24,29)(26,32);(1,4)(2,10)(3,13)(5,20)(6,9)(7,26)(8,22)(11,27)(12,19)(14,29)(15,24)(16,30)(17,31)(18,32)(21,25)(23,28)Polycyclic presentation:
2^9|a^2=g,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,f^c=fi,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 | #72213538, #72267983, C2×#147757, … |
| maximal normal subgroup of | #72213538, #72267983, C2×#147757, … |
| maximal quotient of | C2×#147757 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54953);