Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14519 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,3,12,10)(4,13,16,25,17,8,22,26)(5,18,20,9,21,19,23,14)(15,28,29,24,30,32,31,27);(1,3)(2,7)(4,8)(5,19)(6,10)(9,23)(11,12)(13,17)(14,20)(15,28)(16,26)(18,21)(22,25)(24,29)(27,31)(30,32);(1,4,7,17)(2,8)(3,13)(5,15,21,30)(6,22,12,16)(9,24)(10,26)(11,25)(14,27)(18,32)(19,28)(20,31,23,29);(1,5)(2,9)(3,14)(4,15)(6,23)(7,21)(8,24)(10,19)(11,18)(12,20)(13,27)(16,31)(17,30)(22,29)(25,32)(26,28)Polycyclic presentation:
2^9|a^2=h,c^2=eh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319366, #72319370, #72319444 |
| maximal normal subgroup of | #72319366, #72319370, #72319444 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295592);