Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359665 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13867 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3867 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)(3,13,16,22)(4,17)(5,8)(6,11)(7,12)(9,15,24,21)(10,18)(14,28,31,23)(19,25)(20,26)(27,30,32,29);(2,8)(5,6)(9,22)(10,17)(11,12)(13,24)(15,21)(18,20)(23,32)(25,26)(27,28)(29,30);(1,3)(2,9)(4,14)(5,15)(6,21)(7,16)(8,13)(10,23)(11,22)(12,24)(17,27)(18,29)(19,31)(20,30)(25,32)(26,28);(1,4)(2,10)(3,14)(5,20)(6,18)(7,19)(8,17)(9,23)(11,25)(12,26)(13,27)(15,30)(16,31)(21,29)(22,32)(24,28)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72233773, #72233775, #72233779, … |
| maximal normal subgroup of | #72233773, #72233775, #72233779, … |
| maximal quotient of | #72238621, #72249697, #72251314, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266861);