Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501188 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24404 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #23882 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1103 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)(3,8)(4,14,16,7)(5,9)(6,10)(11,23)(12,29,15,20)(13,24)(17,25,31,21)(18,27)(19,26)(22,28,32,30);(2,7)(3,11)(5,18)(8,22)(9,25)(10,14)(12,28)(13,19)(15,30)(17,31)(20,23)(21,27)(24,32)(26,29);(1,3,6,13)(2,8,10,24)(4,15,16,12)(5,19,18,11)(7,20,14,29)(9,26,27,23)(17,28,31,30)(21,32,25,22);(2,7)(8,20)(9,21)(10,14)(22,23)(24,29)(25,27)(26,32)Polycyclic presentation:
2^9|a^2=f,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,f^b=fh,g^b=gi,e^c=ehi,g^c=gi,f^d=fh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237523, #72237525, #72237544, … |
| maximal normal subgroup of | #72237523, #72237525, #72237544, … |
| maximal quotient of | #72312408, #72312478, #72312591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408386);