Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #374018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14134 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,12,6,4)(3,13,16,9,17,24,22,14)(5,18,19,10,21,26,20,25)(15,23,30,32,31,28,29,27);(1,3)(2,9)(4,13)(5,15)(6,22)(7,16)(8,17)(10,23)(11,24)(12,14)(18,27)(19,30)(20,29)(21,31)(25,28)(26,32);(1,4,7,12,8,11,6,2)(3,14,16,24,17,9,22,13)(5,18,19,25,21,26,20,10)(15,28,30,32,31,23,29,27);(1,5)(2,10)(3,15)(4,18)(6,20)(7,19)(8,21)(9,23)(11,26)(12,25)(13,27)(14,28)(16,30)(17,31)(22,29)(24,32)Polycyclic presentation:
2^9|a^2=h,c^2=fh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215968, #72216144, #72219686, … |
| maximal normal subgroup of | #72215968, #72216144, #72219686, … |
| maximal quotient of | C2×#374020 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281216);