Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511835 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32898 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2371 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)(3,12)(4,18)(5,9)(6,14)(7,13)(8,15)(10,17)(11,20)(16,25)(19,32)(21,29)(22,26)(23,28)(24,30)(27,31);(2,9)(3,5)(10,14)(11,24)(12,15)(13,26)(16,19)(17,22)(18,32)(25,29)(27,28)(30,31);(1,3)(2,10)(4,16)(5,8)(6,22)(7,17)(9,13)(11,25)(12,14)(15,26)(18,24)(19,21)(20,31)(23,30)(27,29)(28,32);(1,4)(2,11)(3,16)(5,19)(6,20)(7,23)(8,21)(9,24)(10,25)(12,27)(13,18)(14,29)(15,28)(17,30)(22,31)(26,32)Polycyclic presentation:
2^9|e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=ci,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,g^d=gi,f^e=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72260522, #72283303, #72340414 |
| maximal normal subgroup of | #72260522, #72283303, #72340414 |
| maximal quotient of | #72261198, #72310908, #72340421 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419031);