Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454087 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16249 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(5,17)(8,23)(9,26)(10,18)(11,29)(12,19)(15,20)(21,25)(22,28)(24,31)(27,32);(1,2)(3,8)(4,9)(5,18)(6,10)(7,17)(11,24)(12,21)(13,25)(14,31)(15,22)(16,28)(19,23)(20,26)(27,30)(29,32);(1,3)(2,8)(4,14)(5,19)(6,13)(7,21)(9,27)(10,25)(11,15)(12,17)(16,30)(18,23)(20,29)(22,32)(24,26)(28,31);(1,4)(2,9)(3,11)(5,20)(6,16)(7,22)(8,24)(10,28)(12,30)(13,29)(14,19)(15,17)(18,26)(21,27)(23,31)(25,32)Polycyclic presentation:
2^9|e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72215974, #72219683, #72219689, … |
| maximal normal subgroup of | #72215974, #72219683, #72219689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361283);