Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419458 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2709 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15375 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1211 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)(4,17)(5,10)(6,23)(7,12)(8,15)(9,20)(11,21)(14,28)(16,30)(18,31)(19,27)(22,29)(24,26)(25,32);(1,3)(2,8)(4,18)(5,19)(6,24)(7,16)(9,10)(11,30)(12,26)(13,23)(14,20)(15,21)(17,29)(22,32)(25,28)(27,31);(1,4)(2,9)(3,14)(5,15)(6,19)(7,20)(8,10)(11,27)(12,17)(13,28)(16,18)(21,32)(22,24)(23,25)(26,29)(30,31);(1,5,21,18,7,22,6,14)(2,10,23,28,12,29,11,31)(3,4,15,32,16,20,24,19)(8,25,13,17,26,27,30,9)Polycyclic presentation:
2^9|d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319954, #72320009, #72320075, … |
| maximal normal subgroup of | #72319954, #72320009, #72320075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335066);