Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493635 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20609 |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #63017 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2269 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,8,18,23)(3,11,14,28,13,27,31,32)(4,15,17,26,16,9,19,24)(7,20,22,12,21,29,25,30);(1,3,6,13)(2,7,8,21)(4,12,16,30)(5,14,18,31)(9,11,15,27)(10,25,23,22)(17,29,19,20)(24,32,26,28);(9,15)(11,27)(24,26)(28,32);(2,8)(5,18)(7,21)(9,15)(11,27)(14,31)(17,19)(20,29)Polycyclic presentation:
2^9|a^2=g,b^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,f^b=fhi,g^b=gh,e^c=eh,g^d=gi,f^e=fi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72232878, #72260522, #72303256 |
| maximal normal subgroup of | #72232878, #72260522, #72303256 |
| maximal quotient of | #72232800, #72233296, #72238425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400838);