Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511825 |
| 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×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2426 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)(4,9)(5,8)(6,11)(7,14)(10,18)(13,26)(15,25)(16,23)(17,22)(19,28)(20,30)(21,32)(24,31)(27,29);(1,3)(2,7)(4,14)(5,15)(6,13)(8,20)(9,12)(10,21)(11,19)(16,28)(17,30)(18,29)(22,25)(23,26)(24,27)(31,32);(3,13)(7,19)(9,23)(14,28)(15,29)(20,21)(22,24)(30,32);(2,8)(5,18)(7,20)(9,22)(10,11)(12,25)(15,29)(17,31)(19,21)(23,24)(26,27)(30,32)Polycyclic presentation:
2^9|g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^a=hi,c^b=ci,f^b=fhi,e^c=eh,g^d=gi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72234103, #72234211, #72234215, … |
| maximal normal subgroup of | #72234103, #72234211, C2×#511825, … |
| maximal quotient of | #72214565, #72219502, C2×#511825, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419021);