Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14694 |
| 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 | |
3645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,11,7,3)(4,13,17,24,16,26,22,9)(5,18,21,10,19,14,20,25)(15,23,29,31,30,32,28,27);(1,3,6,11,8,12,7,2)(4,13,22,24,16,26,17,9)(5,14,20,25,19,18,21,10)(15,27,29,32,30,31,28,23);(1,4)(2,9)(3,13)(5,15)(6,22)(7,17)(8,16)(10,23)(11,24)(12,26)(14,27)(18,31)(19,30)(20,29)(21,28)(25,32);(1,5)(2,10)(3,14)(4,15)(6,20)(7,21)(8,19)(9,23)(11,25)(12,18)(13,27)(16,30)(17,28)(22,29)(24,32)(26,31)Polycyclic presentation:
2^9|a^2=h,b^2=eh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72241161, #72241162, #72241304, … |
| maximal normal subgroup of | #72241161, #72241162, #72241304, … |
| maximal quotient of | C2×#392714 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299910);