Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13360 |
| 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 | |
1649 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)(3,15)(4,19,9,16)(5,23,25,11)(6,12)(7,28,24,13)(8,14)(10,21,29,26)(17,22)(18,20)(27,32)(30,31);(1,3)(2,9)(4,14)(5,16)(6,17)(7,29)(8,18)(10,24)(11,15)(12,21)(13,22)(19,25)(20,23)(26,32)(27,31)(28,30);(1,4)(2,10)(3,13)(5,20)(6,26)(7,30)(8,9)(11,31)(12,16)(14,29)(15,25)(17,23)(18,28)(19,32)(21,27)(22,24)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^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,e^c=ei,f^c=fi,g^c=gi,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 | C2×#147758, #72267972, #72267973, … |
| maximal normal subgroup of | C2×#147758, #72267972, #72267973, … |
| maximal quotient of | C2×#147758 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54954);