Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493562 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1790 |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #62874 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,13,4,10)(3,9,14,26,16,27,17,28)(5,11,18,29,20,30,21,31)(8,22,25,19,24,32,23,15);(1,3)(2,8)(4,14)(5,15)(6,17)(7,16)(9,11)(10,23)(12,25)(13,24)(18,32)(19,20)(21,22)(26,29)(27,30)(28,31);(1,3)(2,9)(4,17)(5,19)(6,14)(7,16)(8,11)(10,28)(12,26)(13,27)(15,20)(18,32)(21,22)(23,31)(24,30)(25,29);(1,4,7,6)(2,10,13,12)(3,14,16,17)(5,18,20,21)(8,23,24,25)(9,26,27,28)(11,29,30,31)(15,32,19,22)Polycyclic presentation:
2^9|a^2=g,d^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,f^b=fhi,g^b=gh,d^c=di,e^c=eh,f^e=fi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72234210, #72234211, #72234212, … |
| maximal normal subgroup of | #72234210, #72234211, #72234212, … |
| maximal quotient of | #72238624, #72251839, #72312732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400768);