Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2847 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14694 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,14,6,13)(3,15,17,27,18,28,16,9)(4,19,21,10,22,29,20,30)(5,23,25,11,26,31,24,32);(1,3)(2,9,14,27)(4,5)(6,16)(7,17)(8,18)(10,23,30,31)(11,19,32,29)(12,15,13,28)(20,24)(21,25)(22,26);(1,4)(2,10)(3,5)(6,21)(7,20)(8,22)(9,23)(11,28)(12,19)(13,29)(14,30)(15,32)(16,25)(17,24)(18,26)(27,31);(1,5)(2,11)(3,4)(6,25)(7,24)(8,26)(9,19)(10,28)(12,23)(13,31)(14,32)(15,30)(16,21)(17,20)(18,22)(27,29)Polycyclic presentation:
2^9|a^2=h,b^2=eh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72238012, #72241162, #72241172, … |
| maximal normal subgroup of | #72238012, #72241162, #72241172, … |
| maximal quotient of | C2×#392711 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299907);