Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13995 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2248 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)(3,13,16,8)(4,17,19,26)(5,11)(6,12)(7,10)(9,20,27,21)(14,30)(15,28,22,24)(18,23)(25,31)(29,32);(1,3)(2,8)(4,18,19,32)(5,22)(6,15)(7,16)(9,25,27,30)(10,13)(11,28)(12,24)(14,21,31,20)(17,23,26,29);(1,4)(2,9)(3,14)(5,21)(6,20)(7,19)(8,23)(10,27)(11,26)(12,17)(13,29)(15,32)(16,31)(18,22)(24,30)(25,28);(1,3)(2,8)(4,14)(5,22)(6,15)(7,16)(9,23)(10,13)(11,28)(12,24)(17,30)(18,21)(19,31)(20,32)(25,26)(27,29)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215953, #72219684, #72219689, … |
| maximal normal subgroup of | #72215953, #72219684, #72219689, … |
| maximal quotient of | #72343043 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273826);