Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493480 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20585 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #62824 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #3484 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2287 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)(3,11,14,24)(4,8)(5,17,19,9)(6,10)(7,13)(12,25,27,32)(15,28,30,23)(16,22)(18,21)(20,26)(29,31);(1,3,6,14)(2,7,10,21)(4,12,16,27)(5,18,19,13)(8,20,22,31)(9,24,17,11)(15,29,30,26)(23,32,28,25);(8,22)(20,31)(23,28)(25,32);(1,4)(2,8)(3,12)(5,15)(6,16)(7,20)(9,23)(10,22)(11,25)(13,26)(14,27)(17,28)(18,29)(19,30)(21,31)(24,32)Polycyclic presentation:
2^9|a^2=g,b^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,g^b=ghi,d^c=dh,f^d=fi,g^e=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72234419, #72258019, #72258042, … |
| maximal normal subgroup of | #72234419, #72258019, #72258042, … |
| maximal quotient of | #72233187, #72256330, #72258253, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400677);